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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The authors announce a construction of quantum deformations of the Schubert varieties for semisimple algebraic groups (which are either classical or of types \(E_ 6\) or \(G_ 2\)). More precisely, the authors consider the coordinate ring \({\mathbf C}[X]\) of a Schubert variety \(X\) and then give a construction of a quantum analogue \({\mathbf C}_ q[X]\). This is done by means of the socalled admissible quadruples and standard monomials [cf. \textit{V. Lakshmibai} and \textit{K. N. Rajeswari}, Contemp. Math. 88, 449-578 (1989; Zbl 0682.14035)]. quantum deformations; Schubert varieties for semisimple algebraic groups V. Lakshmibai and N. Reshetikhin, ''Quantum deformations of flag and Schubert schemes,''C. R. Acad. Sci. Paris Ser. I Math.,313, No. 3, 121--126 (1991). Quantum groups (quantized enveloping algebras) and related deformations, Grassmannians, Schubert varieties, flag manifolds Quantum deformations of flag and Schubert schemes | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Schubert polynomials were introduced in the context of the geometry of flag varieties. This paper investigates some of the connections not yet understood between several combinatorial structures for the construction of Schubert polynomials; we also present simplifications in some of the existing approaches to this area. We designate certain line diagrams for permutations known as rc-graphs as the main structure. The other structures in the literature we study include: semistandard Young tableaux, Kohnert diagrams, and balanced labelings of the diagram of a permutation. The main tools in our investigation are certain operations on rc-graphs, which correspond to the coplactic operations on tableaux, and thus define a crystal graph structure on rc-graphs; a new definition of these operations is presented. One application of these operations is a straightforward, purely combinatorial proof of a recent formula due to \textit{A. S. Buch, A. Kresch, H. Tamvakis}, and \textit{A. Yong} [Duke Math. J. 122, 125--143 (2004; Zbl 1072.14067)], which expresses Schubert polynomials in terms of products of Schur polynomials. In spite of the fact that it refers to many objects and results related to them, the paper is mostly self-contained. Schubert polynomial; Young tableau; rc-graph; crystal graph; Kohnert diagram Lenart, C.: A unified approach to combinatorial formulas for Schubert polynomials. J. Algebr. Comb. 20, 263--299 (2004) Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Grassmannians, Schubert varieties, flag manifolds, Combinatorial aspects of representation theory A unified approach to combinatorial formulas for Schubert polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In their paper [C. R. Acad. Sci., Paris, Sér. I 304, 209--211 (1987; Zbl 0642.13011)], \textit{W. Kraśkiewicz} and \textit{P. Pragacz} defined certain modules, which we call KP modules, over the upper triangular Lie algebra whose characters are Schubert polynomials. In a previous work the author showed that the tensor product of Kraśkiewicz-Pragacz modules always has KP filtration, i.e. a filtration whose each successive quotients are isomorphic to KP modules. In this paper we explicitly construct such filtrations for certain special cases of these tensor product modules, namely \(S_w \otimes S^d(K^i)\) and \(S_w \otimes \bigwedge^d(K^i)\), corresponding to Pieri and dual Pieri rules for Schubert polynomials. Schubert polynomials; Schubert functors; Kraśkiewicz-Pragacz modules; Schubert calculus Classical problems, Schubert calculus, (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.) Kraśkiewicz-Pragacz modules and Pieri and dual Pieri rules for Schubert polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This rather important paper indicates a precise concrete way to perform computations in the quantum equivariant ``deformation'' of the cohomology ring of \(G(k,n)\), the complex Grassmannian variety parametrizing \(k\)-dimensional vector subspaces of \({\mathbb C}^n\). It relies on the results of another important paper, regarding the same subject, by the same author [Adv. Math. 203, 1--33 (2006; Zbl 1100.14045)]. The usual singular cohomology ring of \(G(k,n)\) is a very well known object, studied since Schubert's time, at the end of the XIX Century. First of all, it is a finite free \({\mathbb Z}\)-module generated by the so-called Schubert cycles. Furthermore, the special Schubert cycles, the Chern classes of the universal quotient bundle over \(G(k,n)\), generates it as a \({\mathbb Z}\)-algebra. Multiplying two Schubert cycles then amounts to know how to multiply a special Schubert cycle with a general one (Pieri's formula) and a way to express any Schubert cycle as an explicit polynomial expression in the special Schubert cycles (Giambelli's formula).
The obvious way to deform the cohomology of a Grassmannian is to consider the cohomology of the total space of a Grassmann bundle, parametrizing \(k\)-planes in the fibers of a rank \(n\) vector bundle, which is a deformation of the cohomology of any fiber of it. In the last few decades, however, other ways to deform the cohomology ring of \(G(k,n)\) have been studied. \textit{E. Witten} [in: Geometry, topology and physics for Raoul Bott. Lectures of a conference in honor of Raoul Bott's 70th birthday, Harvard University, Cambridge, MA, USA 1993. Cambridge, MA: International Press. Conf. Proc. Lect. Notes Geom. Topol. 4, 357--422 (1995; Zbl 0863.53054)], introduced the small quantum deformation of the cohomology ring of the Grassmannian, whose structure constants were first determined by \textit{A. Bertram} [Adv. Math. 128, No. 2, 289--305 (1997; Zbl 0945.14031)]. Finally, \textit{A. Knutson} and \textit{T. Tao} [Duke Math. J. 119, No. 2, 221--260 (2003; Zbl 1064.14063)], studied the equivariant deformation of the cohomology of the Grassmannians via the combinatorics of puzzles.
In the beautiful paper under review the author recovers the quantum and equivariant Schubert calculus within a unified framework. Basing on the algebraic properties of the Schur factorial functions, the author realizes the equivariant quantum cohomology ring in terms of generators and relations and gives an explicit basis of polynomial representatives for the equivariant quantum Schubert classes. An alternative approach is offered by \textit{D. Laksov} [Adv. Math. 217, 1869--1888 (2008; Zbl 1136.14042)], where the author proves that the basic results of equivariant Schubert calculus, the basis theorem, Pieri's formula and Giambelli's formula can be obtained from the corresponding results of a more general and elementary framework, as in [\textit{D. Laksov}, Indiana Univ. Math. J., 56, No. 2, 825--845 (2007; Zbl 1136.14042)], by a change of basis.
The paper is organized as follows. Section 1 is the introduction, where the main results are clearly stated and motivated; Section 2 is a useful and very pleasant review of the algebra of factorial Schur functions. The quantum equivariant cohomology of Grassmannians is treated in Section 3, while the proof of the theorem about the presentation of the quantum equivariant cohomology ring is given in Section 4. Section 5 ends the paper with the discussion and the proof of Giambelli's formula in equivariant quantum cohomology. Giambelli's formulas; quantum equivariant Schubert calculus; factorial Schur functions L.C. Mihalcea, \textit{Giambelli formulae for the equivariant quantum cohomology of the Grassmannian}, \textit{Trans. AMS}\textbf{360} (2008) 2285 [math/0506335]. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Symmetric functions and generalizations, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Grassmannians, Schubert varieties, flag manifolds Giambelli formulae for the equivariant quantum cohomology of the Grassmannian | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials These notes are based upon a series of lectures that both authors had given in a summer school at Thurnau, Germany, held from June 19 to June 23, 1995. The lectures were designed to provide an introduction to the theory of Schubert varieties, at its contemporary state of knowledge, and to the related theory of degeneracy loci of vector bundle morphisms in algebraic geometry. The text under review follows closely the lectures delivered at Thurnau, the notes of which had been circulating, since then, among the community of algebraic geometers, but it has been enhanced, in its present published from, by ten additional appendices and a few up-dating remarks or footnotes. As the authors emphasize in the preface to the book, this text is neither intended to be a textbook, nor a research monograph, nor a survey on the subject. Instead, they have tried to describe what they, in their capacity of being two of the most active and competent researches in this area of algebraic geometry, regard as essential features of the whole complex of topics, each from his own point of view. The outcome is a great, huge panorama of a fascinating subject in both classical and contemporary algebraic geometry. The present text consists of nine chapters, ten appendices to them, and an utmost rich bibliography.
Chapter I starts with the classical origin of the whole subject, that is, with the description of loci of matrices of various ranks. This is followed by discussing classical and modern solutions of these old problems, including the combinatorial framework of Schur functions and Schubert polynomials. Chapter II turns to the modern generalization of the classical background, namely to morphisms of vector bundles over algebraic varieties, their degeneracy loci, and the cohomological invariants of these degeneracy loci. The fundamental case of Grassmannians and flag manifolds, together with the Schubert subvarieties associated with them, is the central topic of this chapter. Chapter III is devoted to the crucial combinatorial tools: the various kinds of symmetric functions such as Schur \(S\)-polynomials, Schur \(Q\)-polynomials, supersymmetric polynomials, and others, together with their fundamental properties and identities. Chapter IV discusses symmetric polynomials supported on degeneracy loci of vector bundle maps. The powerful general technique of Gysin maps is also explained in this chapter, and that for the important special case of Grassmannians and flag manifolds. In addition, chapters III and IV touch upon the problem of determining those polynomials that are universally supported on degeneracy loci with an explicit description of their defining ideals. Chapter V gives an application of the technique described in chapter IV to the problem of computing topological Euler characteristics of degeneracy loci and Brill-Noether loci in Jacobians of curves. The geometry of flag manifolds and determinantal formulas for Schubert varieties in the case of general homogeneous spaces associated with various classical groups are treated in chapters VI-VII. Following the correspondence method described in chapter III, degeneracy loci for generalized vector bundles (over homogeneous spaces) are investigated, too. Chapter VIII provides a particularly important application of the general theory developed in chapter VII, namely the computation of cohomology classes of some Brill-Noether loci in Prym varieties.
Although several further applications and open problems are pointed out in the course of chapters I-VIII, the concluding chapter IX is exclusively devoted to the discussion of a huge variety of other applications, related questions, and more open problems.
The following ten appendices A-J serve the purpose of making the text as self-contained as possible, on the one hand, and of indicating some closely related work that has been done since 1995, on the other hand.
Appendix A provides some background material from general intersection theory and the representation theory of degeneracy loci by symmetric polynomials. Appendix B gives a recent improvement of Fulton's theorem on the characterization of vexillary permutations in terms of degeneracy loci. Appendix C points to the relation between degeneracy loci, Demazure's resolution scheme for singularities, and the so-called Bott-Samelson schemes, just so for the sake of completeness. Appendix D compiles the definition and basic properties of Pfaffians, while appendix \(E\) sketches the relevant background material from the group-theoretic approach to Schubert varieties. Appendix F explains a useful Gysin formula for Grassmannian bundles, and appendix G discusses a general criterion for computing the classes of relative diagonals. A special construction for vector bundles, which is well-known and due to D. Mumford, is explained in appendix H (and used in chapter VIII). Appendix I provides a little bit of the relevant representation theory of groups and the combinatorics of Young tableaux, though this is not needed anywhere in the text. Finally, appendix J points to the very recent developments in quantum cohomology, in particular to the significance of the so-called ``quantum double Schubert polynomials'' introduced by \textit{I. Ciocan-Fontanine} and \textit{W. Fulton} (cf.: ``Quantum double Schubert polynomials'', Inst. Mittag-Leffler Report No. 6 (1996-97). Throughout the entire, highly enlightening and inspiring text, the authors have focused on careful explanations of the treated material, with lots of included examples and hints to the original papers. Proofs are mostly just indicated, but always come with precise references to the original papers. The omittance of technical details is to the benefit of the non-expert reader, because this makes the beauty of the entire panorama drawn here more transparent and enjoyable. It should be mentioned that another beautiful, recent introduction to the topic of Schubert varieties and symmetric polynomials is given by the lecture notes ``Fonctions symétriques, polynômes de Schubert et lieux de dégénérescence'' by \textit{L. Manivel} [Cours Spécialisés, No. 3, Paris (1998; Zbl 0911.14023)]. Schubert varieties; quantum double Schubert polynomials; Schur functions; Schubert polynomials; morphisms of vector bundles; degeneracy loci; Grassmannians; flag manifolds; symmetric functions Fulton, W., Pragacz, P.: Schubert Varieties and Degeneracy Loci. Lecture Notes in Mathematics, vol. 1689. Springer, Berlin (1998) Grassmannians, Schubert varieties, flag manifolds, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Determinantal varieties, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], (Equivariant) Chow groups and rings; motives Schubert varieties and degeneracy loci | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In their work on the infinite flag variety, \textit{T. Lam} et al. [Compos. Math. 157, No. 5, 883--962 (2021; Zbl 07358686)] introduced objects called bumpless pipe dreams and used them to give a formula for double Schubert polynomials. We extend this formula to the setting of K-theory, giving an expression for double Grothendieck polynomials as a sum over a larger class of bumpless pipe dreams. Our proof relies on techniques found in an unpublished manuscript of \textit{A. Lascoux} [``Chern and Yang through ice'', Preprint]. Lascoux showed how to write double Grothendieck polynomials as a sum over alternating sign matrices. We explain how to view the Lam-Lee-Shimozono formula as a disguised special case of Lascoux's alternating sign matrix formula.
\textit{A. Knutson} et al. [J. Reine Angew. Math. 630, 1--31 (2009; Zbl 1169.14033)] gave a tableau formula for vexillary Grothendieck polynomials. We recover this formula by showing vexillary marked bumpless pipe dreams and flagged set-valued tableaux are in weight preserving bijection. Finally, we give a bijection between Hecke bumpless pipe dreams and decreasing tableaux. The restriction of this bijection to Edelman-Greene bumpless pipe dreams solves a problem of Lam, Lee, and Shimozono [loc. cit.]. Grothendieck polynomials; bumpless pipe dreams; alternating sign matrices Combinatorial aspects of algebraic geometry, Symmetric functions and generalizations, Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds, Graph polynomials, Combinatorial aspects of representation theory, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) Bumpless pipe dreams and alternating sign matrices | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The context of the research is as follows. Let \(X\) be one of \(\mathrm{Gr}:=\mathrm{Gr}(k,n)\) the Grassmannian, \(\mathrm{IG}:=\mathrm{IG}(k,2n)\) the symplectic Grassmannian, \(\mathrm{OG} :=\mathrm{OG}(k,2n+1)\) the odd orthogonal Grassmannian. Let \(\mathrm{QH}^*(X)\) be the quantum cohomology ring of \(X\). The minimum positive integer \(d\) such that \(q^d\) appears in the quantum product of two Schubert classes in \(\mathrm{QH}^*(X)\) is previously studied. For \(X = \mathrm{Gr}\), it is computed using Young diagrams by \textit{A. Postnikov} [Duke Math. J. 128, No. 3, 473--509 (2005; Zbl 1081.14070); Proc. Am. Math. Soc. 133, No. 3, 699--709 (2005; Zbl 1051.05078)], \textit{W. Fulton} and \textit{C. Woodward} [J. Algebr. Geom. 13, No. 4, 641--661 (2004; Zbl 1081.14076)].
In this paper, the author gives a formula for the minimum quantum degree in terms of inclusions of Young diagrams. The main result of the paper is:
\begin{itemize}
\item The formula of the minimum quantum degree in the case \(X\) is one of \(\mathrm{IG}(n,2n), \mathrm{OG}(n,2n+1)\) (Theorem 1.5).
\item The formula of the minimum quantum degree in the case \(X\) is one of \(\mathrm{IG}(k,2n+1), \mathrm{OG}(k,2n)\) with \(k<2n\) (Theorem 1.6).
\end{itemize}
The key point used to obtain the main results is studying curve neighborhoods by combinatorial models (certain partitions).
The structure of the paper is as follows. Section 2 prepares background about flag varieties, quantum cohomology, curve neighborhoods, and relation with the minimum quantum degree. Section 3 restates results in [\textit{W. Fulton} and \textit{C. Woodward}, J. Algebr. Geom. 13, No. 4, 641--661 (2004; Zbl 1081.14076); \textit{A. Postnikov}, Duke Math. J. 128, No. 3, 473--509 (2005; Zbl 1081.14070); Proc. Am. Math. Soc. 133, No. 3, 699--709 (2005; Zbl 1051.05078)] and gives proof of Theorem 1.5. Section 4 describes indexing sets for Schubert varieties of \(\mathrm{IG}(k,2n)\) and \(\mathrm{OG}(k,2n+1)\). Section 5 describes combinatorics associated with the curve neighborhoods in Section 3. Section 6 proves Theorem 1.6. Section 7 collects some technical proofs for Section 5. minimum quantum degrees; isotropic Grassmannians; curve neighborhoods; Young diagrams Grassmannians, Schubert varieties, flag manifolds, Combinatorial aspects of algebraic geometry, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Minimum quantum degrees for isotropic Grassmannians in types B and C | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The classical cohomology of the Grassmannian \(\text{Gr}(r, n)\) is generated additively by the Schubert classes; the structure constants for the multiplication are the well studied Littlewood-Richardson coefficients. Multiplication by special Schubert subvarieties, in particular by the unique divisorial class, are determined explicitly by the Pieri formula. Considering the natural action of the \(n\)-dimensional torus \(T\) on \(\text{Gr}(r,n)\), one may study the equivariant cohomology of the Grassmannian, the equivariant Littlewood-Richardson coefficients, and the equivariant Pieri formula. Going further, one may include ``quantum'' corrections to the multiplication in cohomology, given by counts of higher degree holomorphic maps to \(\text{Gr}(r,n)\) with incidence conditions with Schubert cycles. In this fashion, one arrives at the equivariant quantum cohomology of the Grassmannian.
The paper under review presents an explicit equivariant quantum Pieri rule for the quantum multiplication by the equivariant divisorial Schubert class. This could be compared to the non-equivariant case obtained by \textit{A. Bertram} [Adv. Math. 128, No.~2, 289--305 (1997; Zbl 0945.14031)]. Moreover, the author also shows the vanishing of equivariant quantum Littlewood-Richardson coefficients in a certain range, and proves a recursive formula satisfied by these coefficients. Schubert calculus; quantum cohomology; quantum Pieri formula L. Mihalcea, ''Equivariant quantum Schubert calculus,'' Adv. Math., vol. 203, iss. 1, pp. 1-33, 2006. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus Equivariant quantum Schubert calculus | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The principal specialization \(\nu_w=\mathfrak{S}_w(1,\dots,1)\) of the Schubert polynomial at \(w\), which equals the degree of the matrix Schubert variety corresponding to \(w\), has attracted a lot of attention in recent years. In this paper, we show that \(\nu_w\) is bounded below by \(1+p_{132}(w)+p_{1432}(w)\) where \(p_u(w)\) is the number of occurrences of the pattern \(u\) in \(w\), strengthening a previous result by \textit{A. E. Weigandt} [Algebr. Comb. 1, No. 4, 415--423 (2018; Zbl 1397.05205)]. We then make a conjecture relating the principal specialization of Schubert polynomials to pattern containment. Finally, we characterize permutations \(w\) whose RC-graphs are connected by simple ladder moves via pattern avoidance. pattern containment; permutation patterns Combinatorial aspects of algebraic geometry, Classical problems, Schubert calculus Principal specializations of Schubert polynomials and pattern containment | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In this paper, we study the \(T_{\mathrm w}\)-equivariant cohomology of the weighted Grassmannians \(\mathrm{wGr}(d,n)\) introduced by \textit{A. Corti} and \textit{M. Reid} [in: Algebraic geometry. A volume in memory of Paolo Francia. Berlin: de Gruyter. 141--163 (2002; Zbl 1060.14071)], where \(T_{\mathrm w}\) is the \(n\)-dimensional torus that naturally acts on \(\mathrm{wGr}(d,n)\). We introduce the equivariant weighted Schubert classes and, after we show that they form a basis of the equivariant cohomology, we give an explicit formula for the structure constants with respect to this Schubert basis. We also find a linearly independent subset \(\{\mathrm wu_1,\dots,\mathrm wu_{n-1}\}\) of \(\mathrm{Lie}(T_w)^\ast\) such that those structure constants are polynomials in \(\mathrm wu_i\)'s with nonnegative coefficients, up to a permutation on the weights. weighted Grassmannians; equivariant weighted Schubert classes; equivariant cohomology; structure constants Abe, H; Matsumura, T, Equivariant cohomology of weighted Grassmannians and weighted Schubert classes, Int. Math. Res. Not. IMRN, 9, 2499-2524, (2015) Grassmannians, Schubert varieties, flag manifolds, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) Equivariant cohomology of weighted Grassmannians and weighted Schubert classes | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials An important combinatorial result in equivariant cohomology and \(K\)-theory Schubert calculus is represented by the formulas of \textit{S. Billey} [Duke Math. J. 96, No. 1, 205--224 (1999; Zbl 0980.22018)], \textit{W. Graham} [``Equivariant \(K\)-theory and Schubert varieties'' (2002)] and \textit{S. Willems} [Bull. Soc. Math. Fr. 132, No. 4, 569--589 (2004; Zbl 1087.19004)] for the localization of Schubert classes at torus fixed points. These formulas work uniformly in all Lie types, and are based on the concept of a root polynomial. We define formal root polynomials associated with an arbitrary formal group law (and thus a generalized cohomology theory). We use these polynomials to simplify the approach of Billey [loc. cit.] and Graham [loc. cit.] and Willems [loc. cit.], as well as to generalize it to connective \(K\)-theory and elliptic cohomology. Another result is concerned with defining a Schubert basis in elliptic cohomology (i.e., classes independent of a reduced word), using the Kazhdan-Lusztig basis of the corresponding Hecke algebra. Schubert classes; Bott-Samelson classes; elliptic cohomology; root polynomial; Kazhdan-Lusztig basis Lenart, C., Zainoulline, K.: Towards generalized cohomology Schubert calculus via formal root polynomials. arXiv:1408.5952 Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Classical problems, Schubert calculus, Motivic cohomology; motivic homotopy theory, \(K\)-theory and homology; cyclic homology and cohomology, Equivariant \(K\)-theory On Schubert calculus in elliptic cohomology | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Schubert polynomials \(\mathfrak{S}_w\) represent cohomology classes of Schubert cycles in the full flag variety. Their coefficients are nonnegative integers. There exists a number of combinatorial formulas for computing these coefficients. This paper is devoted to the following question: when are all the coefficients of a Schubert polynomial equal to \(0\) or \(1\)? Such polynomials are called zero-one Schubert polynomials.
To answer this question, the authors first make the following observation: if a permutation \(\sigma\in S_m\) is a pattern of \(w\in S_n\), then the Schubert polynomial \(\mathfrak{S}_w\) equals a monomial times \(\mathfrak{S}_\sigma\) plus a polynomial with nonnegative coefficients. Hence the set of 0-1 Schubert polynomials is closed under pattern containment. Using Magyar's orthodontia, an inductive algorithm for computing \(\mathfrak{S}_w\) in terms of the Rothe diagram of \(w\), they describe the set of twelve avoided patterns, and also formulate equivalent confitions for a Schubert polynomial to be 0-1 in terms of Rothe diagrams and orthodontic sequences of permutations.
According to the recent result of the same authors about the supports of Schubert polynomials (see [\textit{A. Fink} et al., Adv. Math. 332, 465--475 (2018; Zbl 1443.05179)]), this implies that each 0-1 Schubert polynomial is equal to the integer transform of a generalized permutahedron. Schubert polynomial; pattern avoidance; Rothe diagram Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds, Combinatorial aspects of algebraic geometry, Permutations, words, matrices Zero-one Schubert polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The following results are presented in this paper:
(1) a quantum (multiplicative) generalization of the Horn conjecture which gives a recursive characterization of the possible eigenvalues of a product of unitary matrices,
(2) the saturation conjecture for the fusion structure coefficients for \(\mathrm{SL}(n)\),
(3) transversality statements for quantum Schubert calculus in any characteristic for the ordinary Grassmannians,
(4) determination of the smallest power of \( q\) in an arbitrary (small quantum) product of Schubert varieties in an ordinary Grassmannian. P. Belkale, Quantum generalization of the Horn conjecture. \textit{J. Amer. Math. Soc}. 21 (2008), 365-408. MR2373354 Zbl 1134.14029 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Algebraic moduli problems, moduli of vector bundles Quantum generalization of the Horn conjecture | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The main purpose of this paper is to show that the multiplication of a Schubert polynomial of finite type \(A\) by a Schur function, which we refer to as Schubert vs. Schur problem, can be understood combinatorially from the multiplication in the space of dual \(k\)-Schur functions. Using earlier work by the second author, we encode both problems by means of quasisymmetric functions. On the Schubert vs. Schur side, we study the poset given by the Bergeron-Sottile's \(r\)-Bruhat order, along with certain operators associated to this order. Then, we connect this poset with a graph on dual \(k\)-Schur functions given by studying the affine grassmannian order of Lam-Lapointe-Morse-Shimozono. Also, we define operators associated to the graph on dual \(k\)-Schur functions which are analogous to the ones given for the Schubert vs. Schur problem. This is the first step of our more general program of showing combinatorially the positivity of the multiplication of a dual \(k\)-Schur function by a Schur function. Bruhat order; Schubert polynomials; \(k\)-Schur functions; Hopf algebras Symmetric functions and generalizations, Hopf algebras and their applications, Classical problems, Schubert calculus Schubert polynomials and \(k\)-Schur functions | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials There is a remarkable formula for the principal specialization of a type A Schubert polynomial as a weighted sum over reduced words. Taking appropriate limits transforms this to an identity for the backstable Schubert polynomials recently introduced by \textit{T. Lam} et al. [``Back stable Schubert calculus'', Preprint, \url{arXiv:1806.11233}]. This note identifies some analogues of the latter formula for principal specializations of Schubert polynomials in classical types B, C, and D. We also describe some more general identities for Grothendieck polynomials. As a related application, we derive a simple proof of a pipe dream formula for involution Grothendieck polynomials. Schubert polynomials; Grothendieck polynomials; Coxeter systems; reduced words Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds Principal specializations of Schubert polynomials in classical types | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The authors study a twisted action of the symmetric group on the cohomology ring \(R\) of the variety of complete flags in a complex vector space \(V\) of dimension \(n\). It is known that \(R\) can be identified with the polynomial ring \(\mathbb{C}[x_1, \dots, x_n]\) in \(n\) variables modulo the ideal \(I^+\) of symmetric polynomials without constant term. The usual basis of \(R\) is given by Schubert polynomials (expressing the Schubert cycles in terms of the codimension one Schubert cycles represented by \(x_1, \dots, x_n\)). The usual Demazure operators \(\partial_i\) act on \(R\) and the authors study the action of the symmetric group on \(R\) given by the operators \(s_i = \sigma_i + \partial_i\), where \(\sigma_i\) denotes a simple transposition (reflection). The algebra generated by \(s_i\) and \(x_i\) is isomorphic to the degenerate affine Hecke algebra \(\mathcal H\) considered by Cherednik. The authors construct certain elements in \(\mathcal H\) (Yang-Baxter operators) and use these to define a bilinear form on \(R\). A distinguished basis with respect to this form is extracted (termed affine Schubert polynomials). This is then applied to computing Schubert expansions of Chern classes. cohomology ring; action of the symmetric group; degenerate affine Hecke algebra; variety of complete flags; Schubert polynomials; Schubert cycles Lascoux A., Leclerc B., Thibon J.-Y.: Twisted action of the symmetric group on the cohomology of the flag manifold. Banach Center Publications, Vol. 36, 1996, pp. 111--124 Grassmannians, Schubert varieties, flag manifolds, Classical real and complex (co)homology in algebraic geometry, Group actions on varieties or schemes (quotients) Twisted action of the symmetric group on the cohomology of a flag manifold | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In this paper the authors study root-theoretic Young diagrams(RYD) to investigate the existence of a Lie-type uniform and nonnegative combinatorial rule for Schubert calculus. They provide some formulas for coadjoint varieties of classical Lie type. This is a simplest case after the (co)minuscule family.
Using classical type rules, as well as results of \textit{P.-E. Chaput} and \textit{N. Perrin} [J. Lie Theory 22, No. 1, 17--80 (2012; Zbl 1244.14036)] in the exceptional types, they suggest a connection between polytopality of the set of nonzero Schubert structure constants and planarity of the diagrams.
The main thesis of this paper is that RYDs provide a simple but uniform combinatorial perspective to discuss such questions mathematically, make precise comparisons, and to measure progress towards a rule (uniform, counting, patchwork, or otherwise). root-theoretic Young diagrams; adjoint varieties; Schubert calculus Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus Root-theoretic Young diagrams and Schubert calculus: planarity and the adjoint varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In this well written and many faceted article several analogs of properties of Schur polynomials are provided for Schubert polynomials.
The author gives three different constructions of Bott-Samelson varieties for a general reductive group with a Borel subgroup, and gives a characterization of the Bott-Samelson variety in terms of certain incidence conditions.
In the case when the group is \(Gl(n)\) the results are given as combinatorial interpretations, and it is shown that in this case the coordinate ring of the Bott-Samelson variety consists of generalized Schur modules. This allows the author to compute the generalized Schur polynomials that are the characters of these modules. Using the same arguments as in an earlier paper [\textit{P. Magyar}, Adv. Math. 134, No. 2, 328-366 (1998; Zbl 0911.14024)] the Borel-Weil theorem is proved, and a version of Demazures character formula is worked out in order to obtain a new expression for generalized Schur polynomials.
The theory is used to compute the Schubert polynomials associated to permutations, the theorem of Kraskiewicz and Pragacz is proved, and three new explicit formulas for Schubert polynomials are given.
The essential ingredient in the combinatorial approach are chamber families used by \textit{A. Berenstein, A. Fomin} and \textit{A. Zelevinsky} [Parametrizations of canonical bases and totally positive matrices, Adv.Math. 122, 49-149 (1996)], associated to reduced decompositions of elements in the Weyl group via its wiring diagram. This is translated into the language of generalized Young diagrams. Schubert polynomials; Weyl character formula; Demazure character formula; Borel-Weil theorem; Bott-Samelson varieties; generalized Schur functions; generalized Schur modules; wiring diagram; chamber set; reductive group; Young diagrams Magyar P., Schubert polynomials and Bott-Samelson varieties, Comment. Math. Helv., 1998, 73(4), 603--636 Grassmannians, Schubert varieties, flag manifolds, Combinatorial aspects of representation theory, Representations of quivers and partially ordered sets, Classical problems, Schubert calculus Schubert polynomials and Bott-Samelson varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We give a dozen formulas concerning Schubert and Grothendieck polynomials, and their interrelations, half of them being new, and most of them interesting. In particular, we describe explicitly the decomposition of Schubert polynomials as positive sums of Grothendieck polynomials, and we show that non-commutative Schubert polynomials are obtained by reading the columns of a two-dimensional Cauchy kernel. A six pages summary in English has been added. Schubert polynomials; Grothendieck polynomials Lascoux, A., Schubert & Grothendieck: un bilan bidécennal, Sém. Lothar. Combin., 50, (2003/04) Symmetric functions and generalizations, Classical problems, Schubert calculus, Representations of finite symmetric groups Schubert and Grothendieck: a bidecennial balance | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The question of when two skew Young diagrams produce the same skew Schur function has been well studied. We investigate the same question in the case of stable Grothendieck polynomials, which are the \(K\)-theoretic analogues of the Schur functions. We prove a necessary condition for two skew shapes to give rise to the same dual stable Grothendieck polynomial. We also provide a necessary and sufficient condition in the case where the two skew shapes are ribbons. symmetric functions; Grothendieck polynomials Symmetric functions and generalizations, Combinatorial aspects of representation theory, Classical problems, Schubert calculus Coincidences among skew stable and dual stable Grothendieck polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In a long series of interesting and sometimes deep articles the authors have exploited the properties of the cohomology ring and Grothendieck ring of flag manifolds. The present article extends formulas for Schur functions, used to prove that the representation ring of the symmetric group is a Hopf algebra, to formulas for (what the authors call) Schubert and Grothendieck polynomials, that are generalizations of Schur functions. As a result of their formulas they obtain a beautiful formula concerning reduced representations of the symmetric group.
Unfortunately their work is now so far developed in terminology and notation that it is hard for non-experts to read it. In addition, their presentation is extremely condensed and computational. Perhaps time has come to collect their contributions in a leisurely written monograph? Why not a successor to Macdonald's book on symmetric polynomials [\textit{I. G. Macdonald}, ''Symmetric functions and Hall polynomials'' (1979; Zbl 0487.20007)]? Schubert polynomial; Grothendieck polynomial; cohomology ring of flag manifold; Grothendieck ring of flag manifolds; Hopf algebra; representations of the symmetric group Lascoux, Alain; Schützenberger, Marcel-Paul, Structure de Hopf de l'anneau de cohomologie et de l'anneau de Grothendieck d'une variété de drapeaux, C. R. Acad. Sci. Paris Sér. I Math., 295, 11, 629-633, (1982) Grassmannians, Schubert varieties, flag manifolds, Classical real and complex (co)homology in algebraic geometry, Representations of finite symmetric groups, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry Structure de Hopf de l'anneau de cohomologie et de l'anneau de Grothendieck d'une variété de drapeaux | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials There is a surprising number of fascinating connections between combinatorics, algebra and geometry. Among the most beautiful illustrations of such ties are the relations between symmetric functions, and in particular \textit{Schur polynomials}, on the one hand, the \textit{representation theory} of the \textit{symmetric}- and \textit{general linear} groups, on the other-, and the theory of \textit{Schubert polynomials}, and the geometry and cohomology theory of \textit{flag varieties} and \textit{Schubert varieties}, on the third hand. These connections have intrigued mathematicians for more than hundred years, and have proved extremely fruitful for the development of the fields. The area is still active and expanding. New surprising ties appear amazingly often, and there remain many interesting problems and conjectures. This book contains an introduction to the basic material of the fields and their interrelations. It offers a nice complement to the books ``Symmetric functions and Hall polynomials'' (1998; Zbl 0899.05068), and the notes on ``Schubert polynomials'', Lond. Math. Soc. Lect. Note Ser. 166, 73-99 (1991; Zbl 0784.05061) by \textit{I. G. Macdonald}, and to \textit{W. Fulton}'s book ``Young tableaux. With applications to representation theory and geometry'' [Lond. Math. Soc. Student Texts. 35 (1997; Zbl 0878.14034)]. At many places the book gives a different point of view and chooses different techniques. Is is written in a clear and quick style. Sometimes so quick that more examples would have been useful. The book consists of three separate parts:
Symmetric functions and Schur polynomials.
Schubert polynomials.
The geometry of Schubert varieties and the cohomology of flags varieties.
The contents is roughly as follows:
In the first part the Schur functions are introduced in the original way of C. Jacobi and interpreted in terms of Young tableaux. The Pieri formula, the Jacobi-Trudi formulas, and Giambelli's formula are proved in the usual way. Combinatorial correspondences like those of Knuth, Schensted and Robinson are explained, and the Plactic monoid is defined. Using these tools the Littlewood-Richardson rule for multiplication of Schur polynomials is given. The presentation is an alternative to that used by Macdonald in the book on symmetric functions and Hall polynomials mentioned above.
Several applications of the theory are mentioned and the important Kostka-Foulkes polynomials are defined and their main properties are proved. In a separate section the classical connection between Schur polynomials and the irreducible characters of the symmetric group is given.
The second part of the book is devoted to Schubert polynomials. There are several possible approaches to Schubert polynomials. Here the author chooses the one proposed by S. Fomin and A. N. Kirillov via the Yang-Baxter equation and the Hecke algebras of the symmetric groups. It is shown how Schubert polynomials can be computed, and the main properties of the Schubert polynomials, like symmetries, the Cauchy formula, bases, interpolation, and specialization are proved. The lattice path method of I. Gessel and G. Viennot is explained and used. An interesting problem is to determine how the Schubert polynomials are multiplied. The partial formulas of Monk and the Pieri Formula for Schubert polynomials are proved.
In the third part of the book Grassmann varieties and their Schubert varieties are introduced and studied. Their coordinate rings are described, and the fundamental properties of the singularities of the Schubert varieties are given. Also the cohomology ring of the Grassmann variety is determined. The well known correspondence between Chern classes and the classes of the special Schubert varieties is described, and the well known Thom-Porteous formula is proved. In order to study degeneracy loci the more general theory of flag varieties and their Schubert varieties is studied, and the cohomology rings of the flag varieties are described. One of the highlights of the book is the study of degeneracy loci of maps between vector bundles, and a proof of the beautiful result of Fulton that gives the relation between the classes of certain degeneracy loci and the Schubert polynomials. Young tableaux; symmetric functions; Schur polynomials; Schubert polynomials; Bruhat order; Hecke algebras; Grassmannians; flag varieties; Schubert varieties; plactic ring; Kostka-Foulkes polynomials; symmetric group; general linear group; singularity; Pieri's formula; Jacobi-Trudi formulas; Giambelli's formula; Yang-Baxter equation; Thom-Porteous formula; degeneracy loci; partitions; Littlewood-Richardson rule; Monk's formula Manivel, Laurent, Symmetric functions, Schubert polynomials and degeneracy loci, SMF/AMS Texts and Monographs 6, viii+167 pp., (2001), American Mathematical Society, Providence, RI; Société Mathématique de France, Paris Grassmannians, Schubert varieties, flag manifolds, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Research exposition (monographs, survey articles) pertaining to combinatorics, Combinatorial aspects of representation theory, Enumerative problems (combinatorial problems) in algebraic geometry, Symmetric functions and generalizations, Representations of finite symmetric groups Symmetric functions, Schubert polynomials and degeneracy loci. Transl. from the French by John R. Swallow | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We address a unification of the Schubert calculus problems solved by \textit{A. S. Buch} [Acta Math. 189, No. 1, 37--78 (2002; Zbl 1090.14015)] and \textit{A. Knutson} and \textit{T. Tao} [Duke Math. J. 119, No. 2, 221--260 (2003; Zbl 1064.14063)]. That is, we prove a combinatorial rule for the structure coefficients in the torus-equivariant \(K\)-theory of Grassmannians with respect to the basis of Schubert structure sheaves. This rule is positive in the sense of \textit{D. Anderson} et al. [J. Eur. Math. Soc. (JEMS) 13, No. 1, 57--84 (2011; Zbl 1213.19003)] and in a stronger form. Our work is based on the combinatorics of genomic tableaux and a generalization of \textit{M.-P. Schützenberger}'s jeu de taquin [Lect. Notes Math. 579, 59--113 (1977; Zbl 0398.05011)]. Using our rule, we deduce the two other combinatorial rules for these coefficients. The first is a conjecture of \textit{H. Thomas} and \textit{A. Yong} [``Equivariant Schubert calculus and jeu de taquin'', Ann. Inst. Fourier (Grenoble) (to appear), \url{arXiv:1207.3209}]. The second (found in a sequel to this paper) is a puzzle rule, resolving a conjecture of Knutson and Vakil from 2005. Olivier Pechenik & Alexander Yong, ``Equivariant \(K\)-theory of Grassmannians'', Forum of Mathematics, Pi5 (2017), Grassmannians, Schubert varieties, flag manifolds, Symmetric functions and generalizations, Combinatorial problems concerning the classical groups [See also 22E45, 33C80] Equivariant \(K\)-theory of Grassmannians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The context of the research is in the theory of double Schubert polynomials (of Billey-Haiman for Lie type C and Ikeda-Mihalcea-Naruse for Lie type B, D) as representative polynomials of equivariant Schubert classes of the equivariant cohomology ring of symplectic (Lie type C) and (odd, even) orthogonal flag manifolds (Lie type B, D). The main results contain:
\begin{itemize}
\item The generators for the kernel of the natural map from the ring of Schubert polynomials to the equivariant cohomology ring of flag manifolds type C (Theorem 1) and type D (Theorem 3).
\item The relations between the double Schubert polynomials of Billey-Haiman and the double theta polynomials of Tamvakis-Wilson (Theorem 2) in working with type C. The relationship between the double Schubert polynomials of Ikeda-Mihalcea-Naruse and the double eta polynomials of Tamvakis (Theorem 4) in working with type D.
\item The results in working with type B are nearly identical to type C through an identity between polynomials in subsection 5.1.
\end{itemize}
These main results are a similar, extended continuation of the program laid out in the work of Lascoux and Schützenberger in [\textit{A. Lascoux} and \textit{P. Pragacz}, Mich. Math. J. 48, 417--441 (2000; Zbl 1003.05106); \textit{A. Lascoux} and \textit{M.-P. Schützenberger}, C. R. Acad. Sci., Paris, Sér. I 294, 447--450 (1982; Zbl 0495.14031); Lect. Notes Math. None, 118--144 (1983; Zbl 0542.14031)] for flag manifolds type A. The method used to obtain the main results included:
\begin{itemize}
\item For generators of the kernel of natural maps (Theorems 1, 3): It used the idea from [\textit{H. Tamvakis}, Math. Ann. 314, No. 4, 641--665 (1999; Zbl 0955.14037), Lemma 1] with the transition equations of [\textit{S. Billey}, Discrete Math. 193, No. 1--3, 69--84 (1998; Zbl 1061.05510); \textit{T. Ikeda} et al., Adv. Math. 226, No. 1, 840--886 (2011; Zbl 1291.05222)] to write the Schubert polynomials in this kernel as an explicit linear combination of these generators.
\item For the relations between the double Schubert polynomials with double theta, eta polynomials (Theorems 2, 4): It is based on the equality of the multi-Schur Pfaffian and its orthogonal analog with certain double Schubert polynomials (Propositions 4, 12).
\end{itemize} Schubert polynomials; theta and eta polynomials; Weyl group invariants; flag manifolds; equivariant cohomology Grassmannians, Schubert varieties, flag manifolds, Symmetric functions and generalizations, Actions of groups on commutative rings; invariant theory, Classical problems, Schubert calculus Schubert polynomials, theta and eta polynomials, and Weyl group invariants | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Ikeda-Mihalcea-Naruse's double Schubert polynomials [\textit{T. Ikeda} et al., Adv. Math. 226, No. 1, 840--886 (2011; Zbl 1291.05222)] represent the equivariant cohomology classes of Schubert varieties in the type \(C\) flag varieties. The goal of this paper is to obtain a new tableau formula of these polynomials associated to \textit{vexillary signed permutations} introduced by Anderson-Fulton. To achieve that goal, we introduce \textit{flagged factorial (Schur) \(Q\)-functions}, combinatorially defined functions in terms of \textit{marked shifted tableaux for flagged strict partitions}, and prove their Schur-Pfaffian formula. As an application, we also obtain a new combinatorial formula of factorial \(Q\)-functions of Ivanov in which \textit{monomials} bijectively corresponds to flagged marked shifted tableaux. Schubert polynomials; tableaux; partitions; Schur-Pfaffians Symmetric functions and generalizations, Combinatorial aspects of representation theory, Grassmannians, Schubert varieties, flag manifolds A tableau formula for vexillary Schubert polynomials in type \(C\) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The complex orthogonal and symplectic groups both act on the complete flag variety with finitely many orbits. We study two families of polynomials introduced by \textit{B. J. Wyser} and \textit{A. Yong} [Sel. Math., New Ser. 20, No. 4, 1083--1110 (2014; Zbl 1303.05212); Transform. Groups 22, No. 1, 267--290 (2017; Zbl 1400.14130)] representing the \(K\)-theory classes of the closures of these orbits. Our polynomials are analogous to the Grothendieck polynomials representing \(K\)-classes of Schubert varieties, and we show that like Grothendieck polynomials, they are uniquely characterized among all polynomials representing the relevant classes by a certain stability property. We show that the same polynomials represent the equivariant \(K\)-classes of symmetric and skew-symmetric analogues of \textit{A. Knutson} and \textit{E. Miller}'s [Ann. Math. (2) 161, No. 3, 1245--1318 (2005; Zbl 1089.14007)] matrix Schubert varieties. We derive explicit expressions for these polynomials in special cases, including a Pfaffian formula relying on a more general degeneracy locus formula of \textit{D. Anderson} [Adv. Math. 350, 440--485 (2019; Zbl 1426.14014)]. Finally, we show that taking an appropriate limit of our representatives recovers the \(K\)-theoretic Schur \(Q\)-functions of \textit{T. Ikeda} and \textit{H. Naruse} [Adv. Math. 243, 22--66 (2013; Zbl 1278.05240)]. Schubert calculus; spherical orbits; Grothendieck polynomials; \(K\)-theory; degeneracy loci \(K\)-theory in geometry, Combinatorial aspects of groups and algebras, Applications of methods of algebraic \(K\)-theory in algebraic geometry, Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds \(K\)-theory formulas for orthogonal and symplectic orbit closures | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The authors study the Hilbert--Samuel multiplicity for points of Schubert varieties in the complete flag variety via Gröbner degenerations of the Kazhdan--Lusztig ideal. It is an open problem to give a manifestly positive combinatorial rule for the multiplicity of a Schubert variety \(X_w\) at its torus fixed points \(e_v\in X_w\). In the special case of Grassmannians, this problem was solved about 10 years ago.
The author's approach to the problem runs as follows. A neighborhood of \(e_v \in X_w\) is encoded by the Kazhdan--Lusztig variety \({\mathcal N}_{v,w}\). It is proposed to study a term order \(\prec_{v,w,\pi}\) that depends on \(v, w\) and a shuffling (total ordering) of variables \(\pi\). The corresponding Gröbner degenerations break \({\mathcal N}_{v,w}\), and its tangent cone, into an initial scheme \(\text{{init}}_{\prec_{v,w,\pi}} {\mathcal N}_{v,w}\) whose reduced scheme structure is a union of coordinate subspaces. By construction, multiplicity is the degree of this monomial ideal. It is known that the limit is set-theoretically equidimensional. The authors conjecture that (1) there exists \(\pi\) such that \(\text{init}_{\prec_{v,w,\pi}} {\mathcal N}_{v,w}\) is reduced; and (2) one can choose \(\pi\) such that the corresponding Stanley--Reisner simplicial complex is homeomorphic to a shellable ball or sphere.
The main result of this article proves these conjectures for the covexillary Schubert varieties, i.e., those \(X_w\), where \(w\) avoids the pattern \(3421\). For covexillary Schubert varieties, the key observation is that one can pick \(\pi\) (depending on \(v,w\)) such that the limit scheme is (after \(\pi\)-shuffling the coordinates and crossing by affine space) the limit scheme of a matrix Schubert variety for another covexillary permutation. This yields an explicit Gröbner basis, with squarefree initial terms, for the Kazhdan--Lusztig ideal under \(\prec_{v,w,\pi}\). The authors prove that the limit is reduced, its Stanley--Reisner simplicial complex is homeomorphic to a shellable ball or sphere, and the multiplicity counts the number of facets of this complex. They also obtain a formula for the Hilbert series of the local ring.
This extends work of \textit{V. Lakshmibai} and \textit{J. Weyman} [Adv. Math. 84, No. 2, 179--208 (1990; Zbl 0729.14037)], \textit{J. Rosenthal} and \textit{A. Zelevinsky} [J. Algebr. Comb. 13, 213--218 (2001; Zbl 1015.14025)], \textit{C. Krattenthaler} [Sémin. Lothar. Comb. 45, B45c, 11 p. (2000; Zbl 0965.14023)], \textit{V. Kodiyalam} and \textit{K. N. Raghavan} [J. Algebra 270, No. 1, 28--54 (2003; Zbl 1083.14056)], \textit{V. Kreiman} and \textit{V. Lakshmibai} [Multiplicities of singular points in Schubert varieties of Grassmannians. Berlin: Springer, 553--563 (2003; Zbl 1092.14060)], \textit{T. Ikeda} and \textit{H. Naruse} [Trans. Am. Math. Soc. 361, No. 10, 5193--5221 (2009; Zbl 1229.05287)] and \textit{A. Woo} and \textit{A. Yong} [A Gröbner basis for Kazhdan-Lusztig ideals, preprint (2009), \url{arXiv:0909.0564}]. Schubert varieties; Hilbert-Samuel multiplicities; Gröbner basis L. Li, A. Yong, Some degenerations of Kazhdan-Lusztig ideals and multiplicities of Schubert varieties, Adv. Math. 229 (2012), 633--667. Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus, Combinatorial problems concerning the classical groups [See also 22E45, 33C80] Some degenerations of Kazhdan-Lusztig ideals and multiplicities of Schubert varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The dual stable Grothendieck polynomials are a deformation of the Schur functions, originating in the study of the \(K\)-theory of the Grassmannian. We generalize these polynomials by introducing a countable family of additional parameters such that the generalization still defines symmetric functions. We outline two self-contained proofs of this fact, one of which constructs a family of involutions on the set of reverse plane partitions generalizing the Bender-Knuth involutions on semistandard tableaux, whereas the other classifies the structure of reverse plane partitions with entries 1 and 2. symmetric functions; reverse plane partitions; Bender-Knuth involutions Symmetric functions and generalizations, Combinatorial aspects of representation theory, Partitions of sets, Grassmannians, Schubert varieties, flag manifolds Refined dual stable Grothendieck polynomials and generalized Bender-Knuth involutions | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In analogy with transition equations for type A Schubert polynomials given by Lascoux and Schützenberger (1982), we give recursive formulas for computing representatives of the Schubert classes for the isotropic flag manifolds. These representatives are exactly the Schubert polynomials found in Billey and Haiman (1995). This new approach to finding Schubert polynomials is very closely related to the geometry of the flag manifold and has the advantage that it does not require explicit computations with divided difference operators. The generalized transition equations also lead to a recursion for Stanley symmetric functions and a new proof of Chevalley's intersection formula for Schubert varieties. The proofs involve a careful study of the Bruhat order for the Weyl groups and two simple lemmas for applying divided difference operators. Schubert polynomial; Schubert classes; Stanley symmetric functions; Schubert varieties; Bruhat order; Weyl groups; divided difference operators Sara Billey, Transition equations for isotropic flag manifolds, Discrete Math. 193 (1998), no. 1-3, 69 -- 84. Selected papers in honor of Adriano Garsia (Taormina, 1994). Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Grassmannians, Schubert varieties, flag manifolds Transition equations for isotropic flag manifolds. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We prove two lemmata about Schubert calculus on generalized flag manifolds \(G/B\), and in the case of the ordinary flag manifold \(GL_n/B\) we interpret them combinatorially in terms of descents, and geometrically in terms of missing subspaces. One of them gives a symmetry of Schubert calculus that we christen descent-cycling. Computer experiment shows these two lemmata are surprisingly powerful: they already suffice to determine all of \(GL_n\) Schubert calculus through \(n=5\), and 99.97\%+ at \(n=6\). We use them to give a quick proof of Monk's rule. The lemmata also hold in equivariant (''double'') Schubert calculus for Kac-Moody groups \(G\). Knutson, A.: Descent-cycling in Schubert calculus. Experiment. math. 10, 345-353 (2001) Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds Descent-cycling in Schubert calculus | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We introduce a family of tableaux that simultaneously generalizes the tableaux used to characterize Grothendieck polynomials and \(k\)-Schur functions. We prove that the polynomials drawn from these tableaux are the affine Grothendieck polynomials and \(k-K\)-Schur functions -- Schubert representatives for the \(K\)-theory of affine Grassmannians and their dual in the nil Hecke ring. We prove a number of combinatorial properties including Pieri rules. tableaux; Grothendieck polynomials; \(k\)-Schur functions; affine Grassmannian J. Morse. ''Combinatorics of the K-theory of affine Grassmannians''. Adv. Math. 229 (2012), pp. 2950--2984.DOI. Symmetric functions and generalizations, Combinatorial aspects of representation theory, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Infinite-dimensional Lie (super)algebras Combinatorics of the \(K\)-theory of affine grassmannians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The aim of the paper under review is to show how the quantum Chevalley formula for \(G/B\), as stated by \textit{D. Peterson} [Lectures on quantum cohomology of \(G/B\), MIT (1996)] and proved rigurously by \textit{W. Fulton} and \textit{C. Woodward} [On quantum product of Schubert classes, Preprint, \texttt{http://arxiv.org/abs/math.AG/0112183}], combined with some ideas of \textit{S. Fomin}, \textit{S. I. Gelfand} and \textit{A. Postnikov} [J. Am. Math. Soc. 10, 565--596 (1997; Zbl 0912.14018)], leads to a formula which describes polynomial representatives of the Schubert cohomology classes in the canonical presentation of \(QH^*(G/B)\) in terms of generators and relations. The formula obtained generalizes some results of Fomin, Gelfand and Postnikov from the above mentioned paper. quantum cohomology ring; Grassmann varieties; Schubert cycles Augustin-Liviu Mare, Polynomial representatives of Schubert classes in \?\?*(\?/\?), Math. Res. Lett. 9 (2002), no. 5-6, 757 -- 769. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds Polynomial representatives of Schubert classes in \(QH^*(G/B)\). | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The paper under review investigates the small quantum cohomology ring of general flag varieties \(G/P\). The quantum product deforms the classical cup product by adding contributions from the count of degree \(d\) rational curves on \(G/P\) with prescribed incidence conditions. The authors determine the smallest power of the quantum parameter that can occur in a product of two Schubert classes. This minimal degree is described combinatorially in terms of the Bruhat ordering, and geometrically by the \(11\) equivalent conditions of theorem \(9.1\) in the paper. The classical Chevalley's formula computes the the product of two Schubert classes, one of of them being of codimension \(1\). The methods of this paper allow for a proof of the quantum version of this formula (theorem \(10.1\)). Gromov Witten invariants W. Fulton and C. Woodward, On the quantum product of Schubert classes, \textit{J. Alge-} \textit{braic Geom.}, 13(2004), No.4, 641-661. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds On the quantum product of Schubert classes | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The study of Schubert polynomials is an important and interesting subject in algebraic combinatorics. One of the possible methods for studying Schubert polynomials is through the modules introduced by \textit{W. Kraskiewicz} and \textit{P. Pragacz} [C. R. Acad. Sci., Paris, Sér. I 304, 209--211 (1987; Zbl 0642.13011)]. In this paper the authors show that any tensor product of Kraśkiewicz-Pragacz modules admits a filtration by Kraśkiewicz-Pragacz modules. This result can be seen as a module-theoretic counterpart of a classical result that the product of Schubert polynomials is a positive sum of Schubert polynomials, and gives a new proof to this classical fact. Schubert polynomials; Schubert functors; Kraśkiewicz-Pragacz modules; Schubert calculus Watanabe, M.: Tensor product of kraśkiewicz-pragacz modules. J. algebra 443, 422-429 (2015) Classical problems, Schubert calculus Tensor product of Kraśkiewicz-Pragacz modules | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The quantum cohomology ring of the Grassmannian is determined by the quantum Pieri rule for multiplying by Schubert classes indexed by row or column-shaped partitions. We provide a direct equivariant generalization of Postnikov's quantum Pieri rule for the Grassmannian in terms of cylindric shapes, complementing related work of \textit{V. Gorbounov} and \textit{C. Korff} [Adv. Math. 313, 282--356 (2017; Zbl 1386.14181)] in quantum integrable systems. The equivariant terms in our Graham-positive rule simply encode the positions of all possible addable boxes within one cylindric skew diagram. As such, unlike the earlier equivariant quantum Pieri rule of \textit{Y. Huang} and \textit{C. Li} [J. Algebra 441, 21--56 (2015; Zbl 1349.14173)] and known equivariant quantum Littlewood-Richardson rules, our formula does not require any calculations in a different Grassmannian or two-step flag variety. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds An equivariant quantum Pieri rule for the Grassmannian on cylindric shapes | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials If one starts with a simply laced Dynkin diagram, then a quiver representation has finitely many orbits. Any fixed such orbit determines its equivariant fundamental class, or, its quiver polynomial. These quiver polynomials are universal polynomials representing degeneracy loci. They generalize several important polynomials in algebraic combinatorics (e.g., Giambelli-Thom-Porteous formulas, Schur and Schubert polynomials of Schubert calculus, and the quantum and universal Schubert polynomials). They have several nice structure properties (stability, positivity). The article provides a nonconventional generating description of these polynomials. quivers; quiver varieties; quiver representations, quiver polynomials; Thom polynomials; simply laced Dynkin graphs; degeneracy loci; Schur polynomials Rimányi, R., Quiver polynomials in iterated residue form, J. Algebraic Combin., 40, 2, 527-542, (2014) Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Representations of quivers and partially ordered sets, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Global theory of complex singularities; cohomological properties, Grassmannians, Schubert varieties, flag manifolds Quiver polynomials in iterated residue form | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In this paper, we provide explicit formula for the dual Schubert polynomials of a special class of permutations using certain involution principals on RC-graphs, resolving a conjecture by \textit{A. Postnikov} and \textit{R. P. Stanley} [J. Algebr. Comb. 29, No. 2, 133--174 (2009; Zbl 1238.14036)]. Schubert polynomial; dual Schubert polynomial; Bruhat chains Classical problems, Schubert calculus, Symmetric functions and generalizations, Combinatorial aspects of algebraic geometry An involution on RC-graphs and a conjecture on dual Schubert polynomials by Postnikov and Stanley | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials It is well known that a Schur function is the `limit' of a sequence of Schur polynomials in an increasing number of variables, and that Schubert polynomials generalize Schur polynomials. We show that the set of Schubert polynomials can be organized into sequences, whose `limits' we call Schubert functions. A graded version of these Schubert functions can be computed effectively by the application of mixed shift/multiplication operators to the sequence of variables \(x=(x_1,x_2,x_3,\dots)\). This generalizes the Baxter operator approach to graded Schur functions of G. P. Thomas, and allows the easy introduction of skew Schubert polynomials and functions. Since the computation of these operator formulas relies basically on the knowledge of the set of reduced words of permutations, it seems natural that in turn the number of reduced words of a permutation can be determined with the help of Schubert functions: we describe new algebraic formulas and a combinatorial procedure, which allow the effective determination of the number of reduced words for an arbitrary permutation in terms of Schubert polynomials. Schur function; Schubert polynomials; Schur polynomials; Schubert functions; Baxter operator; reduced words of permutations Winkel, R.: Schubert functions and the number of reduced words of permutations, Sém. lothar. Combin. 39, 1-28 (1997) Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Reflection and Coxeter groups (group-theoretic aspects), Grassmannians, Schubert varieties, flag manifolds Schubert functions and the number of reduced words of permutations | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We construct the Schubert basis of the torus-equivariant \(K\)-homology of the affine Grassmannian of a simple algebraic group \(G\), using the \(K\)-theoretic nil Hecke ring of \textit{B. Kostant} and \textit{S. Kumar} [J. Differ. Geom. 32, No. 2, 549--603 (1990; Zbl 0731.55005)]. This is the \(K\)-theoretic analogue of a construction of D. Peterson in equivariant homology. For the case where \(G=\text{SL}_n\), the \(K\)-homology of the affine Grassmannian is identified with a sub-Hopf algebra of the ring of symmetric functions. The Schubert basis is represented by inhomogeneous symmetric functions, called \(K\)-\(k\)-Schur functions, whose highest-degree term is a \(k\)-Schur function. The dual basis in \(K\)-cohomology is given by the affine stable Grothendieck polynomials, verifying a conjecture of \textit{T. Lam} [J. Am. Math. Soc. 21, No. 1, 259--281 (2008; Zbl 1149.05045)]. In addition, we give a Pieri rule in \(K\)-homology. Many of our constructions have geometric interpretations by means of Kashiwara's thick affine flag manifold. affine Grassmannian; \(K\)-theory; Schubert calculus; symmetric functions; GKM condition Lam, T.; Schilling, A.; Shimozono, M., \textit{K}-theory Schubert calculus of the affine Grassmannian, Compos. Math., 146, 4, 811-852, (2010) Classical problems, Schubert calculus, Symmetric functions and generalizations, \(K\)-theory of schemes \(K\)-theory Schubert calculus of the affine Grassmannian | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The paper under review aims to give an overview of the current state of knowledge about the small quantum cohomology ring of homogeneous varieties, with an emphasis on combinatorial aspects of the theory. Let \(X=G/P\) where \(G\) is a simple complex algebraic Lie group, and \(P\) a parabolic subgroup. The author first summarizes classical results about \(H^{*}(X,\mathbb{Z})\) and explains the relation of this ring to Young tableaux when \(X\) is a Grassmannian. Next, the small quantum cohomology ring \(QH^{*}(X)\) is defined. The author then proceeds to explain some results about the ring structure of \(QH^{*}(X)\). In particular, suppose that \(\sigma_{\lambda}* \sigma_{\mu}=\sum c_{\lambda \mu}^{\nu}(d)q^{d}\sigma_{\nu}\) are the structural equations of the ring in terms of Schubert cycles. Then some of the results discussed in the paper are the quantum Giambelli formula, the computation of \(c_{\lambda\mu}^{\nu}(d)\) in the case of a Grassmannian using classical intersection theory, the smallest value of \(d\) such that \(q^{d}\) appears in the product, and the determination of all \(q^{d}\) which occur with a nonzero coefficient. The paper ends with some remarks and questions. homogeneous spaces; Young tableaux W. Fulton, On the quantum cohomology of homogeneous varieties, The legacy of Niels Henrik Abel, Springer, Berlin, 2004, pp. 729 -- 736. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds On the quantum cohomology of homogeneous varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We establish the formula for multiplication by the class of a special Schubert variety in the integral cohomology ring of the flag manifold. This formula also describes the multiplication of a Schubert polynomial by either an elementary or a complete symmetric polynomial. Thus, we generalize the classical Pieri formula for Schur polynomials (associated to Grassmann varieties) to Schubert polynomials (associated to flag manifolds). Our primary technique is an explicit geometric description of certain intersections of Schubert varieties. This method allows us to compute additional structure constants for the cohomology ring, some of which we express in terms of paths in the Bruhat order on the symmetric group, which in turn yields an enumerative result about the Bruhat order. multiplication by the class of a special Schubert variety; integral cohomology ring of the flag manifold; Pieri formual; Bruhat order Frank Sottile, Pieri's formula for flag manifolds and Schubert polynomials, Ann. Inst. Fourier (Grenoble) 46 (1996), no. 1, 89-110 (English, with English and French summaries). Grassmannians, Schubert varieties, flag manifolds, Symmetric functions and generalizations Pieri's formula for flag manifolds and Schubert polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We give a combinatorial Chevalley formula for an arbitrary weight, in the torus-equivariant \(K\)-group of semi-infinite flag manifolds, which is expressed in terms of the quantum alcove model. As an application, we prove the Chevalley formula for anti-dominant fundamental weights in the (small) torus-equivariant quantum \(K\)-theory \(QK_T(G/B)\) of the flag manifold \(G/B\); this has been a longstanding conjecture. We also discuss the Chevalley formula for partial flag manifolds \(G/P\). Moreover, in type \(A_{n-1}\), we prove that the so-called quantum Grothendieck polynomials indeed represent Schubert classes in the (non-equivariant) quantum \(K\)-theory \(QK (SL_n/B)\). semi-infinite flag manifold; Chevalley formula; quantum Bruhat graph; quantum LS paths; quantum alcove model Combinatorial aspects of representation theory, Combinatorial aspects of algebraic geometry, Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus, Quantum groups and related algebraic methods applied to problems in quantum theory A combinatorial Chevalley formula for semi-infinite flag manifolds and its applications | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In \textit{T. Tao} et al. [J. Am. Math Soc. 17, No. 1, 19--48 (2001; Zbl 1043.05111)] the authors introduced a new rule called a puzzle rule for computing Schubert calculus. In this paper the authors give an independent and nearly self-contained proof of the puzzle rule, they also give a formula for equivariant Schubert calculus on Grassmannians that is manifestly positive in the sense of \textit{W. Graham} [Duke Math. J. 109, 599--614 (2001; Zbl 1069.14055)]. In section 1 the authors state their main result as theorem 2. In section 2, the authors give a combinatorial definition of the equivariant cohomology ring of Gr\(_k(\mathbb C^n)\) stated in section 2.1. In section 2.2 they introduce well known facts about T-equivariant cohomology to apply it in section 2.3 to construct equivariant Schubert classes. In section 2.4 they show that the equivariant Schubert classes form a basis for the equivariant cohomology ring of Gr\(_k(\mathbb C^n)\), concluding in lemma 2 with some properties of the structure constants. In section 3 they state in proposition 2 an equivariant version of the Pieri rule and in theorem 3 they provide a recurrence relation on the structure constants. In corollary 1 and lemma 4 of the same section they introduce four identities numbered (1)-(4). In section 4 they prove indentities (1) and (4). In section 5 they prove identity (3). Finally in section 6 they compare their results with those of \textit{A. I. Molev} and \textit{B. Z. Sagar} [Trans. Am. Math. Soc. 351, 4429--4443 (1999; Zbl 0972.05053)] for multiplying factorial Schur functions (which are equivariant Schubert polynomials for Grassmannian permutations). They introduce cohomological formulations of their problem and a reformulation of their rule in terms of ``Molev-Sagan puzzles''. In the appendix to this paper they extend the standard combinatorial proof of the existence of Schubert classes to equivariant Schubert classes. classical problems, Schubert calculus, groups acting on specific manifolds, equivariant algebraic topology of manifolds Allen Knutson & Terence Tao, ``Puzzles and (equivariant) cohomology of Grassmannians'', Duke Math. J.119 (2003) no. 2, p. 221-260 Classical problems, Schubert calculus, Groups acting on specific manifolds, Equivariant algebraic topology of manifolds Puzzles and (equivariant) cohomology of Grassmannians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials A ``flip-and-reversal'' involution arising in the study of quasisymmetric Schur functions provides a passage between what we term ``Young'' and ``reverse'' variants of bases of polynomials or quasisymmetric functions. Building on this perspective, which has found recent application in the study of \(q\)-analogues of combinatorial Hopf algebras and generalizations of dual immaculate functions, we develop and explore Young analogues of well-known bases for polynomials. We prove several combinatorial formulas for the Young analogue of the key polynomials, show that they form the generating functions for left keys, and provide a representation-theoretic interpretation of Young key polynomials as traces on certain modules. We also give combinatorial formulas for the Young analogues of Schubert polynomials, including their crystal graph structure. We moreover determine the intersections of (reverse) bases and their Young counterparts, further clarifying their relationships to one another. quasisymmetric Schur functions; Schubert polynomials Symmetric functions and generalizations, Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds The ``Young'' and ``reverse'' dichotomy of polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(X_w\) be the Schubert subvariety of the complete flag variety associated to a permutation \(w\) in the symmetric group \(S_n\). It is important to know which \(X_w\) are singular and where \(X_w\) is singular. The authors survey many recent results concerning this problems. They introduce a new combinatorial notion, a generalization of pattern avoidance, which they call interval pattern avoidance, and use this to explore the singularities of Schubert varieties and their local invariants.
In the last sections of the paper a computation approach to the problems based on the Macaulay 2 is discussed. The associated commutative algebra is that of Kazhdan-Lusztig ideals (a class of ideals generalizing classical determinantal ideals). The authors wrote the Macaulay 2 code Schubsingular as an exploratory complement to this paper. It is available at the authors' websites. Schubert varieties; singularities; Kazhdan-Lusztig polynomials; determinantal ideals A. Woo, A. Yong, Governing singularities of Schubert varieties, J. Algebra 320 (2008), no. 2, 495--520. Grassmannians, Schubert varieties, flag manifolds, Singularities in algebraic geometry Governing singularities of Schubert varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The authors obtain the quantum cohomology ring of the flag manifold of type \(G_{2}\). The corresponding quantum Schubert polynomials are explicitly computed. The authors utilize the moment graph of such manifold to calculate all the curve neighborhoods of Schubert classes. Curve neighborhoods method is used to write down Chevalley formula for class multiplication. quantum cohomology; Schubert polynomial; \(G_2\) flag manifold; Chevalley formulas; moment graph Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Combinatorial problems concerning the classical groups [See also 22E45, 33C80] Quantum Schubert polynomials for the \(G_2\) flag manifold | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This book presents the current state of the art and ongoing work on \textit{affine Schubert calculus} with an accent on the combinatorics of a family of polynomials called \(k\)-Schur functions. Several generalizations of \(k\)-Schur functions are also discussed.
In [Duke Math. J. 116, No. 1, 103--146 (2003; Zbl 1020.05069)], \textit{L. Lapointe} et al. found computational evidence of a conjectural property for a family of new bases for a filtration on the symmetric function space: the property is that Macdonald polynomials expand positively in terms of it (see Section 4.11). This gave rise to \(k\)-Schur functions, which then were proven to be connected to a vast set of subjects, see the introduction of the book.
Chapter 2 (which occupies 2/3 of the book) presents basics on \(k\)-Schur functions, emphasizing combinatoric aspects in the symmetric function setting. In particular, \(k\)-Pieri rule for the product of \(k\)-Schur functions is discussed. Also, \(k\)-Schur functions (resp. their duals) generate so called \textit{strong} (resp., \textit{weak}) tableaux, which is explained by means of an affine insertion algorithm. A lot of example in \textit{Sage} is given, the authors hope that this will encourage the reader to generate new data and new conjectures.
Chapter 3 explains the combinatorial connections between Stanley symmetric functions (appeared when Stanley was enumerating reduced words in the symmetric group) and \(k\)-Schur functions, using root systems, nilCoxeter and nilHecke rings. There are exercises in this chapter. Several geometric interpretations of the material are listed at the end of this chapter.
\textit{T. Lam} showed [Am. J. Math. 128, No. 6, 1553--1586 (2006; Zbl 1107.05095)] that the dual \(k\)-Schur functions are a special case of affine analogs of Stanley symmetric functions. Then, the way how Stanley symmetric functions are related to nilCoxeter algebra [\textit{S. Fomin} and \textit{R. P. Stanley}, Adv. Math. 103, No. 2, 196--207 (1994; Zbl 0809.05091)] can be reproduced in the affine setting.
Chapter 4 presents the nilHecke ring in the general Kac-Moody setting, and then this ideology is applied for affine Grassmannians. The nilHecke ring was introduced to study the torus equivariant cohomology of Kac-Moody partial flag varieties, and so this chapter presents this geometric aspect of the story. The algebraic part of correspondence between polynomial representatives for the Schubert classes of the affine Grassmannian, and \(k\)-Schur functions in homology and the dual \(k\)-Schur functions in cohomology is presented. Schur functions; affine Schubert calculus; \(k\)-Schur functions; Macdonald positivity; Pieri rule; nilCoxeter ring; nilHecke ring; Kac-Moody variety Lam, Thomas; Lapointe, Luc; Morse, Jennifer; Schilling, Anne; Shimozono, Mark; Zabrocki, Mike, \(k\)-Schur functions and affine Schubert calculus, Fields Institute Monographs 33, viii+219 pp., (2014), Springer, New York; Fields Institute for Research in Mathematical Sciences, Toronto, ON Research exposition (monographs, survey articles) pertaining to algebraic geometry, Classical problems, Schubert calculus, Homogeneous spaces and generalizations, Symmetric functions and generalizations, Combinatorial aspects of representation theory, Kac-Moody groups, Basic orthogonal polynomials and functions associated with root systems (Macdonald polynomials, etc.) \(k\)-Schur functions and affine Schubert calculus | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We show a \(\mathbb{Z}^2\)-filtered algebraic structure and a ``quantum to classical'' principle on the torus-equivariant quantum cohomology of a complete flag variety of general Lie type, generalizing earlier works of Leung and the second author. We also provide various applications on equivariant quantum Schubert calculus, including an equivariant quantum Pieri rule for partial flag variety \(F \ell_{n_1, \ldots, n_k; n + 1}\) of Lie type \(A\). equivariant quantum cohomology; flag varieties; quantum to classical principle Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds On equivariant quantum Schubert calculus for \(G/P\) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We introduce new notions in elliptic Schubert calculus: the (twisted) Borisov-Libgober classes of Schubert varieties in general homogeneous spaces \(G/P\). While these classes do not depend on any choice, they depend on a set of new variables. For the definition of our classes we calculate multiplicities of some divisors in Schubert varieties, which were only known for full flag varieties before. Our approach leads to a simple recursions for the elliptic classes. Comparing this recursion with R-matrix recursions of the so-called elliptic weight functions of Rimanyi-Tarasov-Varchenko we prove that weight functions represent elliptic classes of Schubert varieties. Grassmannians, Schubert varieties, flag manifolds, Elliptic cohomology, Equivariant \(K\)-theory Elliptic classes of Schubert varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We prove an explicit combinatorial formula for the structure constants of the Grothendieck ring of a Grassmann variety with respect to its basis of Schubert structure sheaves. We furthermore relate \(K\)-theory of Grassmannians to a bialgebra of stable Grothendieck polynomials, which is a \(K\)-theory parallel of the ring of symmetric functions. Buch, A. S., A Littlewood-Richardson rule for the \textit{K}-theory of Grassmannians, Acta Math., 189, 37-78, (2002) Grassmannians, Schubert varieties, flag manifolds, Symmetric functions and generalizations, Combinatorial problems concerning the classical groups [See also 22E45, 33C80] A Littlewood-Richardson rule for the \(K\)-theory of Grassmannians. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We give combinatorial descriptions of the restrictions to \( T\)-fixed points of the classes of structure sheaves of Schubert varieties in the \( T\)-equivariant \( K\)-theory of Grassmannians and of maximal isotropic Grassmannians of orthogonal and symplectic types. We also give formulas, based on these descriptions, for the Hilbert series and Hilbert polynomials at \( T\)-fixed points of the corresponding Schubert varieties.
These descriptions and formulas are given in terms of two equivalent combinatorial models: excited Young diagrams and set-valued tableaux. The restriction formulas are positive, in that for a Schubert variety of codimension \( d\), the formula equals \( (-1)^d\) times a sum, with nonnegative coefficients, of monomials in the expressions \( (e^{-\alpha } -1)\), as \( \alpha \) runs over the positive roots. In types \( A_n\) and \( C_n\) the restriction formulas had been proved earlier by \textit{V. Kreiman} [``Schubert classes in the equivariant \(K\)-theory and equivariant cohomology of the Grassmannian'', Preprint, \url{arXiv:math/0512204}; ``Schubert classes in the equivariant \(K\)-theory and equivariant cohomology of the Lagrangian Grassmannian'', Preprint, \url{arXiv:math/0602245}] using a different method. In type \( A_n\), the formula for the Hilbert series had been proved earlier by \textit{L. Li} and \textit{A.Yong} [Adv. Math. 229, No. 1, 633--667 (2012; Zbl 1232.14033)].
The method of this paper, which relies on a restriction formula of \textit{W. Graham} [``Equivariant \(K\)-theory and Schubert varieties'', Preprint] and \textit{M. Willems} [Duke Math. J. 132, No. 2, 271--309 (2006; Zbl 1118.19002)], is based on the method used by \textit{T. Ikeda} and \textit{H. Naruse} [Trans. Am. Math. Soc. 361, No. 10, 5193--5221 (2009; Zbl 1229.05287)] to obtain the analogous formulas in equivariant cohomology. The formulas we give differ from the \( K\)-theoretic restriction formulas given by \textit{T. Ikeda} and \textit{H. Naruse} [Adv. Math. 243, 22--66 (2013; Zbl 1278.05240)], which use different versions of excited Young diagrams and set-valued tableaux. We also give Hilbert series and Hilbert polynomial formulas which are valid for Schubert varieties in any cominuscule flag variety, in terms of the 0-Hecke algebra. set-valued tableaux W. Graham and V. Kreiman, \textit{Excited Young diagrams, equivariant K-theory, and Schubert varieties}, Trans. AMS, 367 (2015), pp. 6597--6645. Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Grassmannians, Schubert varieties, flag manifolds, Toric varieties, Newton polyhedra, Okounkov bodies, Algebraic combinatorics Excited Young diagrams, equivariant \(K\)-theory, and Schubert varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Based on recent advances on the relation between geometry and representation theory, we propose a new approach to elliptic Schubert calculus. We study the equivariant elliptic characteristic classes of Schubert varieties of the generalized full flag variety \(G/B\). For this first we need to twist the notion of elliptic characteristic class of Borisov-Libgober by a line bundle, and thus allow the elliptic classes to depend on extra variables. Using the Bott-Samelson resolution of Schubert varieties we prove a BGG-type recursion for the elliptic classes, and study the Hecke algebra of our elliptic BGG operators. For \(G = \mathrm{GL}_n (\mathbb{C})\) we find representatives of the elliptic classes of Schubert varieties in natural presentations of the K theory ring of \(G/B\), and identify them with the Tarasov-Varchenko weight function. As a byproduct we find another recursion, different from the known R-matrix recursion for the fixed point restrictions of weight functions. On the other hand the R-matrix recursion generalizes for arbitrary reductive group \(G\). Grassmannians, Schubert varieties, flag manifolds, Elliptic genera, Equivariant \(K\)-theory, Global theory and resolution of singularities (algebro-geometric aspects), Equivariant homology and cohomology in algebraic topology, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry Elliptic classes of Schubert varieties via Bott-Samelson resolution | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This is a survey paper on the quantum cohomology of isotropic Grassmannians. Two questions are investigated here. First, the author explains the quantum Pieri rule, expressing the quantum product of a general Schubert class with a special Schubert class. Secondly, he gives a presentations for the quantum cohomology in terms of generators and relations.
The survey starts with the case of the classical Grassmannians. The quantum Pieri rule was found by \textit{A. Bertram} [Adv. Math. 128, No. 2, 289--305 (1997; Zbl 0945.14031)], while \textit{B. Siebert} and \textit{G. Tian} obtained a presentation of the quantum cohomology [Asian J. Math. 1, No. 4, 679--695 (1997; Zbl 0974.14040)]. The current paper follows the approach of \textit{A. K. Buch, A. Kresch} and \textit{H. Tamvakis} [J. Am. Math. Soc. 16, No. 4, 901--915 (2003; Zbl 1063.53090)], using the idea of kernel and span for a rational map to the Grassmannian to obtain the structure constants for the quantum cohomology. The next section discusses the Lagrangian and maximal isotropic orthogonal Grassmannians, using a similar point of view. Note however that the original arguments in \textit{A. Kresch} and \textit{H. Tamvakis} [J. Algebr. Geom. 12, No. 4, 777--810 (2003; Zbl 1051.53070); Compos. Math. 140, No. 2, 482--500 (2004; Zbl 1077.14083)] made use of Quot scheme compactifications of the moduli space of rational maps to the isotropic Grassmannian. Finally, in the last part of the survey, the author considers the non-maximal isotropic Grassmannians of type B, C, D. Harry Tamvakis, Quantum cohomology of isotropic Grassmannians, Geometric methods in algebra and number theory, Progr. Math., vol. 235, Birkhäuser Boston, Boston, MA, 2005, pp. 311 -- 338. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds Quantum cohomology of isotropic Grassmannians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(V\) be a vector space with a non-degenerate symmetric form and OG be the orthogonal Grassmannian which parametrizes maximal isotropic subspaces in \(V\). The authors give a presentation for the small quantum cohomology ring \(QH^*(\text{OG})\) and show that its product structure is determined by the ring of \(\tilde P\)-polynomials of \textit{P.~Pragacz} and \textit{J.~Ratajski} [Compos. Math. 107, 11--87 (1997; Zbl 0916.14026)]. A ``quantum Schubert calculus'' is formulated, which includes quantum Pieri and Giambelli formulas, as well as algorithms for computing Gromov-Witten invariants. As an application, it is shown that the table of three-point, genus zero Gromov-Witten invariants for OG coincides with that for a corresponding Lagrangian Grassmannian LG, up to an involution.
In a companion paper to this one [\textit{A.~Kresch} and \textit{H.~Tamvakis}, J.~Algebr. Geom., 12, No.~4, 777--810 (2003; Zbl 1051.53070)], the authors provide an analogous analysis for the Lagrangian Grassmannian. The situation in the orthogonal case is similar, but with significant differences, both in the results and in their proofs. quot schemes; Schubert calculus Kresch A., Tamvakis H.: Quantum cohomology of orthogonal Grassmannians. Composit. Math. 140, 482--500 (2004) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds, Combinatorial problems concerning the classical groups [See also 22E45, 33C80] Quantum cohomology of orthogonal Grassmannians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We investigate double transitivity of Galois groups in the classical Schubert calculus on Grassmannians. We show that all Schubert problems on Grassmannians of 2- and 3-planes have doubly transitive Galois groups, as do all Schubert problems involving only special Schubert conditions. We use these results to give a new proof that Schubert problems on Grassmannians of 2-planes have Galois groups that contain the alternating group. We also investigate the Galois group of every Schubert problem on \(\mathrm{Gr}(4,8)\), finding that each Galois group either contains the alternating group or is an imprimitive permutation group and therefore fails to be doubly transitive. These imprimitive examples show that our results are the best possible general results on double transitivity of Schubert problems. Sottile, F.; White, J., Double transitivity of Galois groups in Schubert calculus of Grassmannians, Algebr. Geom., 2, 422-445, (2015) Classical problems, Schubert calculus Double transitivity of Galois groups in Schubert calculus of Grassmannians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In [J. Algebra 448, 238--293 (2016; Zbl 1348.14119)], \textit{A. Yong} and the author study root-theoretic Young diagrams (RYDs), which are one of several natural choices of indexing set for the Schubert present one is that RYDs are useful for studying general patterns in Schubert combinatorics in a uniform manner. The main evidence introduced in that paper is rules for Schubert calculus of the classical (co)adjoint varieties in terms of subvarieties of generalized flag varieties. The thesis of that paper and the RYDs, and a relation between planarity of the root poset for a (co)adjoint variety and polytopalness of the nonzero Schubert structure constants for its cohomology ring. The problem of finding a nonnegative, integral combinatorial rule for the Schubert structure constants of the cohomology ring of a generalized flag variety is longstanding.
In this paper they continue the study of root-theoretic Young diagrams (RYDs). They provide an RYD formula for the \(\mathrm{GL}_n/P\) Belkale-Kumar product, and they give a translation of the indexing set of [\textit{A. S. Buch} et al., Invent. Math. 178, No. 2, 345--405 (2009; Zbl 1193.14071)] for Schubert varieties of non-maximal isotropic Grassmannians into RYDs. Belkale-Kumar product; isotropic Grassmannians; Schubert calculus; adjoint varieties Searles, D., Polynomial bases: positivity and Schur multiplication Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus Root-theoretic Young diagrams and Schubert calculus. II. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials [For the entire collection see Zbl 0665.00004.]
The authors consider a noncommutative version of Schubert polynomials. This is done by simultaneous lifting of the classical Schubert polynomials into two noncommutative algebras related by the ``Dualité de Cauchy''. As a consequence they obtain the functoriality of Schubert polynomials. noncommutative version of Schubert polynomials; functoriality A. Lascoux and M.-P. Schützenberger, Fonctorialité des polynômes de Schubert, Invariant theory (Denton, TX, 1986) Contemp. Math., vol. 88, Amer. Math. Soc., Providence, RI, 1989, pp. 585 -- 598 (French, with English summary). Representations of finite symmetric groups, Grassmannians, Schubert varieties, flag manifolds, Combinatorial identities, bijective combinatorics, Representation theory for linear algebraic groups Fonctorialité des polynômes de Schubert. (Functoriality of Schubert polynomials) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(G\) be a complex reductive algebraic group and \(W\) be its Weyl group. We prove that if \(W\) is of type \(A_{n}\), \(F_{4}\), or \(G_{2}\) and \(w\), \(w'\) are disjoint involutions in \(W\), then the corresponding Kostant-Kumar polynomials do not coincide. As a consequence, we deduce that the tangent cones to the Schubert subvarieties \(X_{w}\), \(X_{w'}\) of the flag variety of \(G\) do not coincide as well. tangent cones; involutions in Weyl groups; Kostant-Kumar polynomials; Schubert varieties Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus Kostant-Kumar polynomials and tangent cones to Schubert varieties for involutions in \(A_{n}\), \(F_{4}\), and \(G_{2}\) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The \(K\)-theory ring of a homogeneous space \(X\) has a natural basis parameterized by the Schubert varieties of \(X\). An important question in enumerative geometry is the determination of the structure constants of the ring with respect to this basis. The paper under review studies the special case when \(X\) is a cominuscule Grassmannian, and one of the multiplied classes corresponds to a special Schubert variety (Pieri rule). The authors describe the structure constants both as integers determined by positive recursive identities and as the number of certain combinatorial objects called tableaux. The result reproves a formula of Lenard in type \(A\), and is new for orthogonal and Lagrangian Grassmannians. The proof is based on calculating sheaf Euler characteristics of special triple intersections of Schubert varieties. Pieri rule; cominuscule Grassmannian; \(K\)-theory Buch, Ander Skovsted; Ravikumar, Vijay, Pieri rules for the \(K\)-theory of cominuscule Grassmannians, J. Reine Angew. Math., 668, 109-132, (2012) Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds, Applications of methods of algebraic \(K\)-theory in algebraic geometry Pieri rules for the \(K\)-theory of cominuscule Grassmannians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Any polynomial in \(\mathbb{Z}[x]\) can be expressed in terms of elementary symmetric polynomials which can in turn be expressed as linear combinations of standard elementary monomials or SEM. The paper under review examines the SEM expansions of Schur and Schubert polynomials. The SEM expansion for Schubert polynomials is of particular interest because quantum Schubert polynomials can be computed by quantizing the SEM expansion of ordinary Schubert polynomials [\textit{S. Fomin}, \textit{S. Gelfand} and \textit{A. Postnikov}, Quantum Schubert polynomials, J. Am. Math. Soc. 10, No. 3, 565-596 (1997; Zbl 0912.14018)].
In the case of Schur functions, the SEM expansion can be obtained via a variant of the Jacobi-Trudi identity, and the author demonstrates how a combinatorial rule based on posets of staircase box diagrams can be used to calculate SEM expansion coefficients.
In the case of Schubert polynomials, the SEM expansion can be obtained via the determinantal expression of \textit{A. N. Kirillov} and \textit{T. Maeno} [Quantum double Schubert polynomials, quantum Schubert polynomials and Vafa-Intriligator formula, q-alg/9610022, 1996, preprint, 52 pp.] and the author conjectures a combinatorial rule similar to that proved for Schur functions. elementary symmetric polynomials; standard elementary monomials; Schubert polynomials; SEM expansion; Schur functions Rudolf Winkel, On the expansion of Schur and Schubert polynomials into standard elementary monomials, Adv. Math. 136 (1998), no. 2, 224-250. Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds On the expansion of Schur and Schubert polynomials into standard elementary monomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials It is well known that many geometric properties of Schubert varieties of type \(A\) (and others) can be interpreted combinatorially. Given two permutations \(w, x \in S_n\) we give a combinatorial consequence of the property that the smooth locus of the Schubert variety \(X_w\) contains the Schubert cell \(Y_x\). This provides a necessary ingredient for the interpretation of recent representation-theoretic results of \textit{E. Lapid} and \textit{A. Mínguez} [Adv. Math. 339, 113--190 (2018; Zbl 1400.20047)] in terms of identities of Kazhdan-Lusztig polynomials. Schubert varieties; permutations Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Grassmannians, Schubert varieties, flag manifolds, Linear algebraic groups over local fields and their integers A tightness property of relatively smooth permutations | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The problem of computing products of Schubert classes in the cohomology ring can be formulated as the problem of expanding skew Schur polynomial into the basis of ordinary Schur polynomials. We reformulate the problem of computing the structure constants of the Grothendieck ring of a Grassmannian variety with respect to its basis of Schubert structure sheaves in a similar way; we address the problem of expanding the generating functions for skew reverse-plane partitions into the basis of polynomials which are Hall-dual to stable Grothendieck polynomials. From this point of view, we produce a chain of bijections leading to Buch's \(K\)-theoretic Littlewood-Richardson rule. Grothendieck polynomial; Littlewood-Richardson rule; tabloid; sign-reversing involution; Yamanouchi word Combinatorial aspects of representation theory, Classical problems, Schubert calculus A dual approach to structure constants for \(K\)-theory of Grassmannians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Given two Schubert classes \(\sigma_{\lambda}\) and \(\sigma_{\mu}\) in the quantum cohomology of a Grassmannian, we construct a partition \(\nu \), depending on \(\lambda\) and \(\mu \), such that \(\sigma_{\nu}\) appears with coefficient 1 in the lowest (or highest) degree part of the quantum product \(\sigma_{\lambda}\bigstar \sigma_{\mu}\). To do this, we show that for any two partitions \(\lambda\) and \(\mu\), contained in a \(k \times (n - k)\) rectangle and such that the \(180^{\circ}\)-rotation of one does not overlap the other, there is a third partition \(\nu\), also contained in the rectangle, such that the Littlewood-Richardson number \(c_{\lambda \mu}^{\nu}\) is 1. quantum cohomology; toric tableau; Littlewood-Richardson number Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds, Combinatorial aspects of representation theory A note on quantum products of Schubert classes in a Grassmannian | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials It is well-established that Schur functions are related to the cohomology of Grassmannians. They can be seen as particular cases of Schubert polynomials, which describe the cohomology of flag varieties. Moreover, Schubert polynomials are generalized by Grothendieck polynomials which are defined via \(K\)-theory rather than cohomology. While semi-standard tableaux give a combinatorial description of Schur functions, \textit{A. S. Buch} [Acta Math. 189, No. 1, 37--78 (2002; Zbl 1090.14015)] has shown that set-valued tableaux give a combinatorial description of (stable) Grothendieck polynomials.
This idea of considering set-valued (rather than integer valued) objects was further extended to the theory of P-partition by \textit{T. Lam} and \textit{P. Pylyavskyy} [Int. Math. Res. Not. 2007, No. 24, Article ID rnm125, 48 p. (2007; Zbl 1134.16017)]. The main goal of the present work is to give an ``enriched'' analog of their results, which means that the underlying combinatorics is similar to that appearing in the theory of Schur \(P\)- and \(Q\)-functions (where two copies of \(\mathbb{N}\) are used as labels in the combinatorial objects). This is motivated by \textit{J. R. Stembridge}'s theory of enriched \(P\)-partitions [Trans. Am. Math. Soc. 349, No. 2, 763--788 (1997; Zbl 0863.06005)].
The authors consider the generating functions of their enriched set-valued \(P\)-partitions (in the same way as Schur functions can be seen as generating functions of semi-standard tableaux). What they obtain are symmetric functions that in some sense generalize \textit{T. Ikeda} and \textit{H. Naruse}'s shifted stable Grothendieck polynomials [Adv. Math. 243, 22--66 (2013; Zbl 1278.05240)]. Along the way, they consider various related Hopf algebras, such as an algebra of labeled posets and some subalgebras of quasisymmetric functions. symmetric functions; quasisymmetric functions; Hopf algebras; posets; Grothendieck polynomials Partitions of sets, Symmetric functions and generalizations, Combinatorial aspects of representation theory, Grassmannians, Schubert varieties, flag manifolds Enriched set-valued \(P\)-partitions and shifted stable Grothendieck polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The \(q\)-Whittaker function \(W_\lambda (\mathfrak{x};q)\) associated to a partition \(\lambda\) is a \(q\)-analogue of the Schur function \(s_\lambda (\mathfrak{x})\), and the \(t=0\) specialization of the Macdonald polynomial \(P_\lambda (\mathfrak{x};q,t)\). We give a new formula for \(W_\lambda (\mathfrak{x};q)\) in terms of partial flags compatible with a nilpotent endomorphism over the finite field of size \(1/q\), analogous to a well-known formula for the Hall-Littlewood functions. We show that considering pairs of partial flags and taking Jordan forms leads to a probabilistic bijection between nonnegative-integer matrices and pairs of semistandard tableaux of the same shape, which we call the \(q\)-Burge correspondence. In the \(q \rightarrow 0\) limit, we recover a description of the classical Burge correspondence (also known as column RSK) due independently to \textit{E. R. Gansner} [SIAM J. Algebraic Discrete Methods 2, 429--440 (1981; Zbl 0498.05038); \textit{N. Spaltenstein}, Classes unipotentes et sous-groupes de Borel. Berlin-Heidelberg-New York: Springer-Verlag (1982; Zbl 0486.20025); \textit{R. Steinberg}, J. Algebra 113, No. 2, 523--528 (1988; Zbl 0653.20039)] for permutation matrices, and to \textit{D. Rosso} [Can. J. Math. 64, No. 5, 1090--1121 (2012; Zbl 1267.14067)] in general. Finally, we apply the \(q\)-Burge correspondence to prove enumerative formulas for certain modules over the preprojective algebra of a path quiver. \(q\)-Whittaker function; finite field; Jordan form; partial flag variety; Burge correspondence; RSK correspondence; preprojective algebra; socle filtration \(q\)-calculus and related topics, Symmetric functions and generalizations, Incomplete beta and gamma functions (error functions, probability integral, Fresnel integrals), Grassmannians, Schubert varieties, flag manifolds \(q\)-Whittaker functions, finite fields, and Jordan forms | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In the paper an explicit formula is proven for the multiplication of an arbitrary Schubert cycle by a special Schubert cycle in the Chow (or cohomology) ring of the homogeneous spaces \(Sp (2m)/P\) and \(SO (2m + 1)/P\), where \(P\) is a maximal parabolic subgroup. These homogeneous spaces are interpreted as Grassmannians of isotropic subspaces of a fixed dimension in \(2m\)-dimensional (resp. \((2m + 1)\)-dimensional) vector space endowed with a non-degenerate symplectic (resp. orthogonal) form. The method follows an earlier paper by the author [Manuscr. Math. 79, No. 2, 127-151 (1993; Zbl 0789.14041)] and uses the divided difference description of Borel's characteristic map in the basis of Schubert cycles given by \textit{I. N. Bernstein}, \textit{I. M. Gel'fand} and \textit{S. I. Gel'fand} [Russ. Math. Surv. 28, No. 3, 1-26 (1973); translation from Usp. Mat. Nauk 28, No. 3(171), 3-26 (1973; Zbl 0286.57025)] and \textit{M. Demazure} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 7, 53-88 (1974; Zbl 0312.14009)]. This allows one to reformulate the original intersection theory problem into some questions of purely algebro-combinatorial nature. As a by-product one obtains some Giambelli-type formulas for these isotropic Grassmannians. Schubert cycle; isotropic Grassmannians Pragacz, P., Ratajski, J.: A Pieri-type theorem for Lagrangian and odd orthogonal Grassmannians. J. Reine Angew. Math. 476, 143--189 (1996) Grassmannians, Schubert varieties, flag manifolds, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Homogeneous spaces and generalizations, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Enumerative problems (combinatorial problems) in algebraic geometry A Pieri-type theorem for Lagrangian and odd orthogonal Grassmannians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Double Kostka polynomials \(K_{\lambda,\mu}(t)\) are polynomials in \(t\), indexed by double partitions \({\lambda,\mu}\). As in the ordinary case, \(K_{\lambda,\mu}(t)\) is defined in terms of Schur functions \(s_\lambda(x)\) and Hall-Littlewood functions \(P_\mu(x;t)\). In this paper, we study combinatorial properties of \(K_{\lambda,\mu}(t)\) and \(P_\mu(x;t)\). In particular, we show that the Lascoux-Schützenberger type formula holds for \(K_{\lambda,\mu}(t)\) in the case where \(\mu = (-,\mu^{\prime\prime})\). Moreover, we show that the Hall bimodule \(\mathscr{M}\) introduced by \textit{M. Finkelberg} et al. [Sel. Math., New Ser. 14, No. 3--4, 607--628 (2009; Zbl 1215.20041)] is isomorphic to the ring of symmetric functions (with two types of variables) and the natural basis \(\mathfrak{u}_\lambda\) of \(\mathscr{M}\) is sent to \(P_\lambda(x;t)\) (up to scalar) under this isomorphism. This gives an alternate approach for their result. Schur functions; Hall-Littlewood functions S. Liu and T. Shoji, Double Kostka polynomials and Hall bimodule, \doihref10.3836/tjm/1475723088Tokyo J. Math., 39 (2017), 743--776. Symmetric functions and generalizations, Combinatorial aspects of representation theory, Representation theory for linear algebraic groups, Reflection and Coxeter groups (group-theoretic aspects), Grassmannians, Schubert varieties, flag manifolds Double Kostka polynomials and Hall bimodule | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We construct the affine version of the Fomin-Kirillov algebra, called the affine FK algebra, to investigate the combinatorics of affine Schubert calculus for type \(A\). We introduce Murnaghan-Nakayama elements and Dunkl elements in the affine FK algebra. We show that they are commutative as Bruhat operators, and the commutative algebra generated by these operators is isomorphic to the cohomology of the affine flag variety. As a byproduct, we obtain Murnaghan-Nakayama rules both for the affine Schubert polynomials and affine Stanley symmetric functions. This enable us to express \(k\)-Schur functions in terms of power sum symmetric functions. We also provide the definition of the affine Schubert polynomials, polynomial representatives of the Schubert basis in the cohomology of the affine flag variety. affine flag variety; affine Fomin-Kirillov algebra; affine nilCoxeter algebra; affine Schubert polynomials; \(k\)-Schur function; Murnaghan-Nakayama rule Combinatorial aspects of algebraic geometry, Classical problems, Schubert calculus Combinatorial description of the cohomology of the affine flag variety | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The aim of this article is to give explicit formulae for various generating functions, including the generating function of torus-invariant primitive ideals in the big cell of the quantum minuscule Grassmannian of type \(B_n\).
Let \(\mathfrak g\) be a simple Lie algebra of rank \(n\) over the field of complex numbers, and let \(\pi:=\{\alpha_1,\ldots,\alpha_n\}\) be the set of simple roots associated to a triangular decomposition \(\mathfrak g=\mathfrak n^-\oplus\mathfrak h\oplus\mathfrak n^+\). Let \(W\) be the Weyl group associated to \(\mathfrak g\).
The aim of this article is to study the prime spectrum of so-called quantum Schubert cells from the point of view of algebraic combinatorics. Quantum Schubert cells have been introduced by \textit{C. De Concini, V. G. Kac} and \textit{C. Procesi} [Stud. Math., Tata Inst. Fundam. Res. 13, 41--65 (1995; Zbl 0878.17014)] as quantisations of enveloping algebras of nilpotent Lie algebras \(\mathfrak n_w:=\mathfrak n^+\cap\mathrm{Ad}_w(\mathfrak n^-)\), where \(\mathrm{Ad}\) stands for the adjoint action and \(w\in W\). These noncommutative algebras are defined thanks to the braid group action of \(W\) on the quantised enveloping algebra \(U_q(\mathfrak g)\) induced by Lusztig automorphisms. The resulting (quantum) algebra associated to a chosen \(w\in W\) is denoted by \(U_q[w]\). Here \(q\) denotes a nonzero element of the base field \(\mathbb K\), and we assume that \(q\) is not a root of unity. It was recently shown by \textit{M. Yakimov} [Proc. Am. Math. Soc. 138, No. 4, 1249--1261 (2010; Zbl 1245.16030)] that these algebras can be seen as the Schubert cells of the quantum flag varieties. Our aim is to study combinatorially the prime spectrum of the algebras \(U_q[w]\). quantum algebras; quantized enveloping algebras; primitive ideals; quantum Schubert cells; quantum flag varieties; algebraic combinatorics Ring-theoretic aspects of quantum groups, Quantum groups (quantized enveloping algebras) and related deformations, Grassmannians, Schubert varieties, flag manifolds, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Ideals in associative algebras Enumeration of torus-invariant strata with respect to dimension in the big cell of the quantum minuscule Grassmannian of type \(B_n\). | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials By applying a Gröbner-Shirshov basis of the symmetric group \(S_n\), we give two formulas for Schubert polynomials, either of which involves only nonnegative monomials. We also prove some combinatorial properties of Schubert polynomials. As applications, we give two algorithms to calculate the structure constants for Schubert polynomials, one of which depends on Monk's formula. divided difference; Schubert polynomial; Gröbner-Shirshov basis Symmetric functions and generalizations, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Grassmannians, Schubert varieties, flag manifolds, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) On formulas and some combinatorial properties of Schubert polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Kazhdan-Lusztig-polynomials \(P_{x,y}(t)\) are associated with two elements \(x,y\) of a given Coxeter group. They find interpretations in combinatorics (poset-recursion), algebra (entries of a transition matrix between two natural bases of the associated Hecke algebra), geometry (Poincaré polynomial of a stalk of the intersection cohomology sheaf on a Schubert variety in the corresponding flag variety), representation theory (the coefficients are the dimensions of certain Ext groups).
The combinatorial definition of Kazhdan-Lusztig polynomials is generalized to more general posets by Stanley, the resulting polynomial is called the Kazhdan-Lusztig-Stanley polynomial. The paper under review is an accessible, beautifully written survey of these polynomials, and their role in geometry. More precisely, consider the following two classes of KLS polynomials: the \(g\)-polynomial of a rational polytope, and the KL polynomial of a hyperplane arrangement. These have geometric interpretations as Poincaré polynomials of stalks of certain intersection cohomology sheafs of an associated geometric object. This result has been known before, but this paper gives a unified proof, which also clarifies which ingredients are needed for the proof.
Moreover, the author introduces a generalization of KLS polynomials called Z-polynomials. They take into account the poset structure as well as the opposite poset structure, or in other words, both right and left KLS polynomials. Geometric interpretations of Z-polynomials analogous to the ones mentioned above for KLS polynomials are proved. Kazhdan-Lusztig; intersection cohomology Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Enumerative problems (combinatorial problems) in algebraic geometry, Classical problems, Schubert calculus The algebraic geometry of Kazhdan-Lusztig-Stanley polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The Schubert calculus on a Grassmannian is a very well known topic. In fact, intersection theory on a Grassmannian is done in an algebraic-combinatorial way via Schur polynomials. On a general flag manifold, a canonical cell decomposition and a duality result for the corresponding closures of the cells (i.e. the Schubert varieties) are known. However there is not much more knowledge about the intersection of arbitrary Schubert varieties. The goal of the paper under review is to study the intersection of Schubert varieties in the varieties of full flags. The main result of the paper, as the author says, may be thought as an analogue of the Pieri formula. Most of the paper is devoted to the algebraic-combinatorial methods developed to prove this key result, which is a first step in the full understanding of the whole intersection theory of flag varieties. Schubert varieties; intersection theory on a Grassmannian; varieties of full flags Grassmannians, Schubert varieties, flag manifolds, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry Intersections of Schubert varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The authors give a description of the intersection cohomology groups of the Schubert varieties in partial flag manifolds over symmetrizable Kac-Moody Lie algebras in terms of parabolic Kazhdan-Lusztig polynomials introduced by \textit{V. Deodhar} [J. Algebra 111, 483-506 (1987; Zbl 0656.22007)]. Grassmanians; Schubert varieties; flag manifolds; intersection cohomology groups; symmetrizable Kac-Moody Lie algebras Kashiwara, M., \& Tanisaki, T. (2002). Parabolic Kazhdan-Lusztig polynomials and Schubert varieties. J. Algebra, 249, 306--325. Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Grassmannians, Schubert varieties, flag manifolds, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) Parabolic Kazhdan-Lusztig polynomials and Schubert varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We study the problem of expanding the product of two Stanley symmetric functions \(F_w \cdot F_u\) into Stanley symmetric functions in some natural way. Our approach is to consider a Stanley symmetric function as a stabilized Schubert polynomial \(F_w = \lim_{n\to \infty} \mathfrak{S}_{1^n\times w}\), and study the behavior of the expansion of \(\mathfrak{S}_{1^n\times w} \cdot \mathfrak{S}_{1^n\times u}\) into Schubert polynomials, as \(n\) increases. We prove that this expansion stabilizes and thus we get a natural expansion for the product of two Stanley symmetric functions. In the case when one permutation is Grassmannian, we have a better understanding of this stability. Stanley symmetric functions; Schubert polynomials; Littlewood-Richardson rule Symmetric functions and generalizations, Classical problems, Schubert calculus A canonical expansion of the product of two Stanley symmetric functions | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Consider the ring \(\mathcal{S}\) of symmetric polynomials in \(k\) variables over an arbitrary base ring \(\mathfrak{k}\). Fix \(k\) scalars \(a_1, a_2, \ldots, a_k\) in \(\mathfrak{k}\). Let \(I\) be the ideal of \(\mathcal{S}\) generated by \(h_{n-k+1}-a_1, h_{n-k+1}-a_2,\ldots, h_{n-k+1}-a_k\), where \(h_i\) is the \(i\)-th complete homogeneous symmetric polynomial.
The quotient ring \(\mathbf{S}/I\) generalizes both the usual and the quantum cohomology of the Grassmannian.
We show that \(\mathbf{S}/I\) has a \(\mathfrak{k}\)-module basis consisting of (residue classes of) Schur polynomials fitting into a \(k \times (n-k)\)-rectangle; and that its multiplicative structure constants satisfy the same \(S_3\)-symmetry as those of the Grassmannian cohomology. We conjecture the existence of a Pieri rule (proven in two particular cases) and a positivity property generalizing that of Gromov-Witten invariants. symmetric functions; partitions; Schur functions; Gröbner bases; Grassmannian; cohomology Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Classical problems, Schubert calculus A quotient of the ring of symmetric functions generalizing quantum cohomology | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We derive explicit formulae for the generating series of mixed Grothendieck dessins d'enfant/monotone/simple Hurwitz numbers, via the semi-infinite wedge formalism. This reveals the strong piecewise polynomiality in the sense of Goulden-Jackson-Vakil, generalising a result of Johnson, and provides a new explicit proof of the piecewise polynomiality of the mixed case. Moreover, we derive wall-crossing formulae for the mixed case. These statements specialise to any of the three types of Hurwitz numbers, and to the mixed case of any pair. Enumerative problems (combinatorial problems) in algebraic geometry, Dessins d'enfants theory, Arithmetic aspects of dessins d'enfants, Belyĭ theory, Coverings of curves, fundamental group, Low-dimensional topology of special (e.g., branched) coverings Wall-crossing formulae and strong piecewise polynomiality for mixed Grothendieck dessins d'enfant, monotone, and double simple Hurwitz numbers | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We establish the relation of Berenstein-Kazhdan's decoration function and Gross-Hacking-Keel-Kontsevich's potential on the open double Bruhat cell in the base affine space \(\text{G} / \mathcal{N}\) of a simple, simply connected, simply laced algebraic group \(G\). As a byproduct we derive explicit identifications of polyhedral parametrization of canonical bases of the ring of regular functions on \(\text{G} / \mathcal{N}\) arising from the tropicalizations of the potential and decoration function with the classical string and Lusztig parametrizations. In the appendix we construct maximal green sequences for the open double Bruhat cell in \(\text{G} / \mathcal{N}\) which is a crucial assumption for Gross-Hacking-Keel-Kontsevich's construction. cluster algebras; quantum groups; canonical bases; mirror symmetry Cluster algebras, Mirror symmetry (algebro-geometric aspects), Quantum groups (quantized enveloping algebras) and related deformations, Quantum groups (quantized function algebras) and their representations Polyhedral parametrizations of canonical bases \& cluster duality | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The author proposes a theory of combinatorially explicit Schubert polynomials that represent the Schubert classes in the Borel presentation of the cohomology ring of the flag variety \({\mathfrak X}\) of the symplectic group \({\text{Sp}}_{2n}\). These polynomials are used to describe the arithmetic Schubert calculus on \({\mathfrak X}\). It is also given a method to compute the natural arithmetic Chern numbers on \({\mathfrak X}\) and to show that they are all rational numbers. flag variety; symplectic group Harry Tamvakis, Schubert polynomials and Arakelov theory of symplectic flag varieties, J. Lond. Math. Soc. (2) 82 (2010), no. 1, 89 -- 109. Grassmannians, Schubert varieties, flag manifolds, Arithmetic varieties and schemes; Arakelov theory; heights, Combinatorial problems concerning the classical groups [See also 22E45, 33C80] Schubert polynomials and Arakelov theory of symplectic flag varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This article deals with the quantum cohomology of the classical flag manifold \(\text{GL}(n,\mathbb C)/B\), where \(B\) is the subgroup of upper triangular nonsingular matrices. The cohomology ring can be obtained as a factor ring of the polynomials in \(n\) variables modulo the ideal generated by the elementary symmetric polynomials. A distinguished basis for the classical cohomology ring is given by the Schubert polynomials. The quantum cohomology ring again is a factor ring of the polynomial ring but now with \(n-1\) additional variables, the deformation parameters, modulo the ideal generated by the ``quantum elementary polynomials''. The author relates the quantum Schubert polynomial and some other related basis with problems in algebraic combinatorics. For the proofs and further details he mainly refers to the following publications of the author [Adv. Math. 136, No. 2, 224-250 (1998; Zbl 0920.05069) and ``On algebraic and combinatorial properties of Schur and Schubert polynomials'' (Bayreuther Math. Schr. 59) (2000; Zbl 0958.05001)]. quantum cohomology; flag manifold; Schubert polynomial; elementary symmetric polynomial; standard elementary monomial Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Grassmannians, Schubert varieties, flag manifolds, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Enumerative problems (combinatorial problems) in algebraic geometry From quantum cohomology to algebraic combinatorics: The example of flag manifolds | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The aim of this paper is to give a review of the main results on Schubert varieties and their generalizations. In the first section Schubert varieties (over \(\mathbb{C})\) are introduced, in the setting of the theory of reductive groups and their Bruhat decomposition. Some geometric results are discussed. The Steinberg variety associated to a reductive group is also introduced. Most of the material of this section is `classical'.
In section 2 examples are given of constructions of algebraic objects, based on the geometry discussed in section 1. For example, 2.2 gives an elementary geometric construction of the Weyl group \(W\) of a reductive group \(G\). It uses correspondences on the flag variety \(X\) of \(G\). Using machinery from algebraic topology, a calculus of correspondences on \(X\) produces the Hecke algebra \({\mathcal H}\) of \(W\). This is discussed in 2.4 and 2.5.
Section 3 discusses generalizations of Schubert varieties. These occur, for example in the context of spherical varieties. A closed subgroup \(H\) of \(G\) is spherical if a Borel subgroup \(B\) of \(G\) has finitely many orbits on \(G/H\). Then \(G/H\) is a (homogeneous) spherical variety. The orbit closures generalize Schubert varieties (which one recovers for \(H=B)\). An important special case is the case of symmetric varieties, where \(H\) is the fixed point group of an involutorial automorphism of \(G\).
The combinatorial properties of the set of orbits are discussed in 3.6. A calculus of correspondences gives rise to a representation of \({\mathcal H}\), discussed in 3.7. The last part of section 3 reviews special features of the case of symmetric varieties. Schubert varieties; reductive group; Bruhat decomposition; Steinberg variety; flag variety; Hecke algebra; spherical varieties Springer, T.A.: Schubert varieties and generalizations. In: \textit{Representation theories and algebraic geometry} (Montreal, PQ, 1997), NATO Adv.~Sci.~Inst.~Ser.~C~Math.~Phys.~Sci., vol 514, Kluwer~Acad.~Publ., Dordrecht, pp. 413-440 (1998) Grassmannians, Schubert varieties, flag manifolds, Homogeneous spaces and generalizations, Linear algebraic groups over the reals, the complexes, the quaternions Schubert varieties and generalizations | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials These are notes for four lectures given at the Osaka summer school on Schubert calculus in (2012), presenting the geometry from the unpublished [the author, ``Puzzles, positroid varieties, and equivariant K-theory of Grassmannians'', Preprint, \url{arXiv:1008.4302}] giving an extension of the puzzle rule for Schubert calculus to equivariant \(K\)-theory, while eliding some of the combinatorial detail. In particular, \S3 includes background material on equivariant cohomology and \(K\)-theory.
Since that school, I have extended the results to arbitrary interval positroid varieties (not just those arising in Vakil's geometric Littlewood-Richardson rule), in the preprint [the author, ``Schubert calculus and shifting of interval positroid varieties'', Preprint, \url{arXiv:1408.1261})]. Classical problems, Schubert calculus, Applications of methods of algebraic \(K\)-theory in algebraic geometry, Grassmannians, Schubert varieties, flag manifolds, Symmetric functions and generalizations Schubert calculus and puzzles | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The main purpose of this paper is to show that the multiplication of a Schubert polynomial of finite type \(A\) by a Schur function can be understood from the multiplication in the space of dual \(k\)-Schur functions. Using earlier work by the second author, we encode both problems by means of quasisymmetric functions. On the Schubert vs. Schur side, we study the \(r\)-Bruhat order given by Bergeron-Sottile, along with certain operators associated to this order. On the other side, we connect this poset with a graph on dual \(k\)-Schur functions given by studying the affine Grassmannian order of Lam-Lapointe-Morse-Shimozono. Also, we define operators associated to the graph on dual \(k\)-Schur functions which are analogous to the ones given for the Schubert vs. Schur problem. Schubert polynomials; \(k\)-Schur functions; affine Grassmannian; \(r\)-Bruhat order; strong order Symmetric functions and generalizations, Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds Schubert polynomials and \(k\)-Schur functions (extended abstract). | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We prove an identity relating the product of two opposite Schubert varieties in the (equivariant) quantum \(K\)-theory ring of a cominuscule flag variety to the minimal degree of a rational curve connecting the Schubert varieties. We deduce that the sum of the structure constants associated to any product of Schubert classes is equal to 1. Equivalently, the sheaf Euler characteristic map extends to a ring homomorphism defined on the quantum \(K\)-theory ring. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), \(K\)-theory of schemes, Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds Euler characteristics of cominuscule quantum \(K\)-theory | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The factorial flagged Grothendieck polynomials are defined by flagged set-valued tableaux of \textit{A. Knutson} et al. [J. Reine Angew. Math. 630, 1--31 (2009; Zbl 1169.14033)]. We show that they can be expressed by a Jacobi-Trudi type determinant formula, generalizing the work of \textit{T. Hudson} and the first author [Eur. J. Comb. 70, 190--201 (2018; Zbl 1408.14030)]. As an application, we obtain alternative proofs of the tableau and the determinant formulas of vexillary double Grothendieck polynomials, which were originally obtained by Knutson et al. [loc. cit.] and Hudson and the first author [loc. cit.] respectively. Furthermore, we show that each factorial flagged Grothendieck polynomial can be obtained by applying \(K\)-theoretic divided difference operators to a product of linear polynomials. factorial Grothendieck polynomials; flagged partitions; flagged set-valued tableaux; vexillary permutations; Jacobi-Trudi formula; double Grothendieck polynomials Classical problems, Schubert calculus, Combinatorial aspects of algebraic geometry Factorial flagged Grothendieck polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In the paper under review, it is shown that quantum homogeneous coordinate rings of generalised flag manifolds corresponding to minuscule weights, their Schubert varieties, big cells, and determinantal varieties are AS-Cohen-Macaulay. The notion of a quantum graded algebra with a straightening law, introduced by [\textit{T. H. Lenagan} and \textit{L. Rigal}, J. Algebra 301, No. 2, 670--702 (2006; Zbl 1108.16026)] is effectively used as a main tool. Using Stanley's Theorem it is moreover shown that quantum generalised flag manifolds of minuscule weight and their big cells are AS-Gorenstein. quantum flag manifolds; straightening laws; Cohen-Macaulay; Gorenstein S. Kolb, The AS-Cohen-Macaulay property for quantum flag manifolds of minuscule weight. J. Algebra 319 (2008), 3518-3534. Quantum groups (quantized enveloping algebras) and related deformations, Rings with straightening laws, Hodge algebras, Grassmannians, Schubert varieties, flag manifolds, Ring-theoretic aspects of quantum groups The AS-Cohen-Macaulay property for quantum flag manifolds of minuscule weight | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This paper surveys a new actively developing direction in contemporary mathematics which connects quantum integrable models with the Schubert calculus for quiver varieties: there is a purely geometric construction of solutions to the Yang-Baxter equation and their associated Yang-Baxter algebras which play a central role in quantum integrable systems and exactly solvable (integrable) lattice models in statistical physics. A simple but explicit example is given using the classical geometry of Grassmannians in order to explain some of the main ideas. The degenerate five-vertex limit of the asymmetric six-vertex model is considered, and its associated Yang-Baxter algebra is identified with a convolution algebra arising from the equivariant Schubert calculus of Grassmannians. It is also shown how our methods can be used to construct quotients of the universal enveloping algebra of the current algebra \(\mathfrak{gl}_2[t]\) (so-called Schur-type algebras) acting on the tensor product of copies of its evaluation representation \(\mathbb{C}^2[t]\). Finally, our construction is connected with the cohomological Hall algebra for the \(A_1\)-quiver. quantum integrable systems; quiver varieties; quantum cohomologies Yang-Baxter equations, Grassmannians, Schubert varieties, flag manifolds, Representations of quivers and partially ordered sets, Groups and algebras in quantum theory and relations with integrable systems Yang-Baxter algebras, convolution algebras, and Grassmannians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Schubitopes were introduced by \textit{C. Monical} et al. [Sel. Math., New Ser. 25, No. 5, Paper No. 66, 37 p. (2019; Zbl 1426.05175)] as a specific family of generalized permutohedra. It was proven by \textit{A. Fink} et al. [Adv. Math. 332, 465--475 (2018; Zbl 1443.05179)] that Schubitopes are the Newton polytopes of the dual characters of flagged Weyl modules. Important cases of Schubitopes include the Newton polytopes of Schubert polynomials and key polynomials. In this paper, we develop a combinatorial rule to generate the vertices of Schubitopes. As an application, we show that the vertices of the Newton polytope of a key polynomial can be generated by permutations in a lower interval in the Bruhat order, settling a conjecture of Monical et al. [loc. cit.]. Schubert polynomial; key polynomial; Newton polytope; Schubitope; Schubert matroid; rank function; Bruhat order; Bruhat order polytope Symmetric functions and generalizations, Combinatorial aspects of representation theory, Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.), Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus Vertices of Schubitopes | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Involution Schubert polynomials represent cohomology classes of \(K\)-orbit closures in the complete flag variety, where \(K\) is the orthogonal or symplectic group. We show they also represent \(\mathsf{T}\) -equivariant cohomology classes of subvarieties defined by upper-left rank conditions in the spaces of symmetric or skew-symmetric matrices. This geometry implies that these polynomials are positive combinations of monomials in the variables \(x_i+x_j\) , and we give explicit formulas of this kind as sums over new objects called involution pipe dreams. Our formulas are analogues of the Billey-Jockusch-Stanley formula for Schubert polynomials. In Knutson and Miller's approach to matrix Schubert varieties, pipe dream formulas reflect Gröbner degenerations of the ideals of those varieties, and we conjecturally identify analogous degenerations in our setting. Schubert polynomials; pipe dreams; involutions; symmetric groups; spherical varieties Grassmannians, Schubert varieties, flag manifolds, Compactifications; symmetric and spherical varieties, Symmetric functions and generalizations Involution pipe dreams | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We give a direct proof of the equivalence between the Giambelli and Pieri type formulas for Hall--Littlewood functions using Young's raising operators, parallel to joint work with Buch and Kresch for the Schubert classes on isotropic Grassmannians. We prove several closely related mirror identities enjoyed by the Giambelli polynomials, which lead to new recursions for Schubert classes. The raising operator approach is applied to obtain tableau formulas for the Hall--Littlewood functions, the theta polynomials of [\textit{A.S. Buch}, \textit{A. Kresch}, and \textit{H. Tamvakis} [``A Giambelli formula for isotropic Grassmannians'', \url{arXiv:0811.2781}], and related Stanley symmetric functions. Finally, we introduce the notion of a skew element \(w\) of the hyperoctahedral group and identify the set of reduced words for \(w\) with the set of standard \(k\)-tableaux on a skew shape \(\lambda/\mu\). Tamvakis H.: Giambelli, Pieri, and tableau formulas via raising operators. J. Reine Angew. Math. 652, 207--244 (2011) Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds, Symmetric functions and generalizations Giambelli, Pieri, and tableau formulas via raising operators | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let K be a compact connected Lie group, T be a maximal torus in K and T' be its normalizer in K. The flag variety \(X=K/T\) admits a cellular decomposition \(X=\cup X_ w\) with cells indexed by all elements w of the Weyl group \(W=T'/T\). The closures \(\bar X{}_ w\) of these cells determine elements in the dual space to the complex cohomology space \(H^*(X)\) that are called Schubert cycles.
The aim of the present paper is to define and to study the analogues of Schubert cycles for the equivariant cohomology \(H^*_ K(X)\). Identifying (by the Chern-Weil isomorphism) \(H^*_ K(X)\) with the space of polynomial functions on the Lie algebra \({\mathfrak t}\) of T the author gives an explicit formula for the equivariant Schubert cycles in terms of the reduced decompositions of elements of W. polynomial functions on Lie algebra; compact connected Lie group; maximal torus; flag variety; cellular decomposition; Weyl group; Schubert cycles; equivariant cohomology; reduced decompositions A. Arabia, Cycles de Schubert et cohomologie équivariante de \?/\?, Invent. Math. 85 (1986), no. 1, 39 -- 52 (French). Discrete subgroups of Lie groups, Homology with local coefficients, equivariant cohomology, Grassmannians, Schubert varieties, flag manifolds Cycles de Schubert et cohomologie équivariante de K/T. (Schubert cycles and equivariant cohomology of K/T) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(\mathfrak g\) be an arbitrary (not necessarily symmetrizable) Kac-Moody Lie algebra with a Cartan subalgebra \(\mathfrak h\), associated group \(G\), Borel subgroup \(B\), maximal torus \(T\) corresponding to \(\mathfrak h\), and Weyl group \(W\). For any \(w\in W\) let \(X_ w\) denote the corresponding Schubert variety in \(G/B\). The author proves in this general setting many properties of Schubert varieties known to hold in the finite case. Among other things he proves that \(X_ w\) is a normal Cohen-Macaulay variety, constructs an explicit rational resolution of \(X_ w\) and proves the Demazure character formula. The method used in the paper seems to give a new proof even in the finite case. In particular it does no rely on any characteristic \(p\) approach.
As a consequence the author gets the Weyl-Kac character formula and the denominator formula for arbitrary Kac-Moody algebras. Kac-Moody Lie algebra; Schubert variety; normal Cohen-Macaulay variety; Weyl-Kac character formula S. Kumar, ''Demazure character formula in arbitrary Kac-Moody setting,'' Invent. Math., vol. 89, iss. 2, pp. 395-423, 1987. Grassmannians, Schubert varieties, flag manifolds, Infinite-dimensional Lie (super)algebras, Infinite-dimensional Lie groups and their Lie algebras: general properties Demazure character formula in arbitrary Kac-Moody setting | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We consider the Grassmannian \(X = G r_{n - k}(\mathbb{C}^n)\) and describe a `mirror dual' Landau-Ginzburg model \((\check{\mathbb{X}}^\circ, W_q : \check{\mathbb{X}}^\circ \to \mathbb{C})\), where \(\check{\mathbb{X}}^\circ\) is the complement of a particular anti-canonical divisor in a Langlands dual Grassmannian \(\check{\mathbb{X}}\), and we express \(W\) succinctly in terms of Plücker coordinates. First of all, we show this Landau-Ginzburg model to be isomorphic to one proposed for homogeneous spaces in a previous work by the second author. Secondly we show it to be a partial compactification of the Landau-Ginzburg model defined in the 1990's by \textit{T. Eguchi} et al. [Int. J. Mod. Phys. A 12, No. 9, 1743--1782 (1997; Zbl 1072.32500)]. Finally we construct inside the Gauss-Manin system associated to \(W_q\) a free submodule which recovers the trivial vector bundle with small Dubrovin connection defined out of Gromov-Witten invariants of \(X\). We also prove a \(T\)-equivariant version of this isomorphism of connections. Our results imply in the case of Grassmannians an integral formula for a solution to the quantum cohomology \(D\)-module of a homogeneous space, which was conjectured by the second author. They also imply a series expansion of the top term in Givental's \(J\)-function, which was conjectured in [\textit{V. V. Batyrev} et al., Nucl. Phys., B 514, No. 3, 640--666 (1998; Zbl 0896.14025)]. mirror symmetry; Gromov-Witten theory; Grassmannian quantum cohomology; cluster algebras; Landau-Ginzburg model; Gauss-Manin system Mirror symmetry (algebro-geometric aspects), Cluster algebras The \(B\)-model connection and mirror symmetry for Grassmannians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The article contains an elementary geometric proof of a Pieri-type formula in the cohomology ring of a grassmannian of maximal isotropic subspaces of orthogonal, or symplectic, vector spaces. This formula was first given by \textit{H. Hiller} and \textit{B. Boe} [Adv. Math. 62, 49-67 (1986; Zbl 0611.14036)], and a different proof was found by \textit{P. Pragacz} and \textit{J. Ratajski} [Manuscr. Math. 79, 127-151 (1993; Zbl 0789.14041)]. In the present article the proof proceeds via explicit computations of triple intersections of Schubert varieties, where one is a special Schubert variety. This method leads to a nice geometric interpretation of the intersection multiplicities as the intersection numbers of linear subspaces coming from the special Schubert varieties with quadrics, and linear subspaces coming from the two other Schubert varieties. Schubert varieties; isotropic grassmannians; triple intersections; Pieri formulas; intersection multiplicities Sottile, F.: Pieri-type formulas for maximal isotropic Grassmannians via triple intersections. Colloq. Math. 82, 49--63 (1999) Grassmannians, Schubert varieties, flag manifolds Pieri-type formulas for maximal isotropic Grassmannians via triple intersections | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Suppose \(K\subseteq\mathrm{GL}(n,\mathbb C)\) is a closed subgroup which acts on the complete flag variety with finitely many orbits. When \(K\) is a Borel subgroup, these orbits are Schubert cells, whose study leads to Schubert polynomials and many connections to type A Coxeter combinatorics. When \(K\) is \(\mathrm{O}(n,\mathbb C)\) or \(\mathrm{Sp}(n,\mathbb C)\), the orbits are indexed by some involutions in the symmetric group. \textit{B. Wyser} and \textit{A. Yong} [Transform. Groups 22, No. 1, 267--290 (2017; Zbl 1400.14130)] described polynomials representing the cohomology classes of the orbit closures, and we investigate parallels for these ``involution Schubert polynomials'' of classical combinatorics surrounding type A Schubert polynomials. We show that their stable versions are Schur-P-positive, and obtain as a byproduct a new Littlewood-Richardson rule for Schur P-functions.
A key tool is an analogue of weak Bruhat order on involutions introduced by \textit{R. W. Richardson} and \textit{T. A. Springer} [Geom. Dedicata 35, No. 1--3, 389--436 (1990; Zbl 0704.20039); complements ibid. 49, No. 2, 231--238 (1994; Zbl 0826.20045)]. This order can be defined for any Coxeter group \(W\), and its labelled maximal chains correspond to reduced words for distinguished elements of \(W\) which we call atoms. In type A we classify all atoms, generalizing work of \textit{M. B. Can}, \textit{M. Joyce} and \textit{B. Wyser} [``Wonderful symmetric varieties and Schubert polynomials'', Ars Math. Contemp. (to appear)], and give a connection to the Chinese monoid of \textit{J. Cassaigne} et al. [Int. J. Algebra Comput. 11, No. 3, 301--334 (2001; Zbl 1024.20046)].We give a different description of some atoms in general finite \(W\) in terms of strong Bruhat order. Schubert polynomials; Coxeter combinatorics; spherical orbits Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Classical problems, Schubert calculus, Reflection and Coxeter groups (group-theoretic aspects), Linear algebraic groups over arbitrary fields Involution Schubert-Coxeter combinatorics | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(V\) be a complex vector space equipped with a nondegenerate symmetric bilinear form. Let \(X\) denote the flag variety for the even orthogonal group, which parameterizes flags of isotropic subspaces in \(V\). In the paper under review, the author develops a theory of combinatorially explicit Schubert polynomials which represent the Schubert classes in the Borel presentation of the cohomology ring of \(X\). These polynomials are applied to understand the structure of the Gillet-Soulé arithmetic Chow ring of \(X\). Actually, the author studies an extra relation which comes from the vanishing of the top Chern class of the maximal isotropic subbundle of the trivial vector bundle over \(X\) and he computes the natural arithmetic Chern numbers on \(X\). Finally, he shows that these arithmetic Chern numbers are all rational. Schubert polynomial; orthogonal flag variety; arithmetic Chow ring; arithmetic Chern number Harry Tamvakis, Schubert polynomials and Arakelov theory of orthogonal flag varieties, Math. Z. 268 (2011), no. 1-2, 355 -- 370. Grassmannians, Schubert varieties, flag manifolds, Arithmetic varieties and schemes; Arakelov theory; heights, Combinatorial problems concerning the classical groups [See also 22E45, 33C80] Schubert polynomials and Arakelov theory of orthogonal flag varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The \(K\)-homology ring of the affine Grassmannian of \(\mathrm{SL}_n(\mathbb{C})\) was studied by Lam, Schilling, and Shimozono [\textit{T. Lam} et al., Compos. Math. 146, No. 4, 811--852 (2010; Zbl 1256.14056)]. It is realized as a certain concrete Hopf subring of the ring of symmetric functions. On the other hand, for the quantum \(K\)-theory of the flag variety \(Fl_n\), \textit{A. N. Kirillov} and \text{T. Maeno} [``A note on quantum \(K\)-theory of flag varieties and some quadric algebras'', in preparation] provided a conjectural presentation based on the results obtained by \textit{A. Givental} and \textit{Y.-P. Lee} [Invent. Math. 151, No. 1, 193--219 (2003; Zbl 1051.14063)]. We construct an explicit birational morphism between the spectrums of these two rings. Our method relies on \textit{S. N. M. Ruijsenaars}'s [Commun. Math. Phys. 133, No. 2, 217--247 (1990; Zbl 0719.58019)] relativistic Toda lattice with unipotent initial condition. From this result, we obtain a \(K\)-theory analogue of the so-called Peterson isomorphism for (co)homology. We provide a conjecture on the detailed relationship between the Schubert bases, and, in particular, we determine the image of Lenart-Maeno's quantum Grothendieck polynomial associated with a Grassmannian permutation. Applications of methods of algebraic \(K\)-theory in algebraic geometry, Hopf algebras and their applications, \(K\)-theory and homology; cyclic homology and cohomology, Relations of \(K\)-theory with cohomology theories, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Quantization in field theory; cohomological methods Peterson isomorphism in \(K\)-theory and relativistic Toda lattice | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The author first introduces \(A\) type double Schubert polynomials, then discusses flagged Schur polynomials, and type \(A\) duality. Finally, he considers the geometry of the single and double Schubert polynomials, focusing on the symplectic case. double Schubert polynomials; flagged Schur polynomials, reverse double Schubert polynomials Classical problems, Schubert calculus, Symmetric functions and generalizations, Combinatorial aspects of representation theory Schubert polynomials and degeneracy locus formulas | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials \textit{C. Gaetz} and \textit{Y. Gao} [Proc. Am. Math. Soc. 148, No. 1, 1--7 (2020; Zbl 07144479)] used an order lowering operator \(\nabla\), introduced by \textit{R. P. Stanley} [Enumerative combinatorics. Vol. 1. 2nd ed. Cambridge: Cambridge University Press (2012; Zbl 1247.05003)], to prove the strong Sperner property for the weak Bruhat order on the symmetric group. Hamaker, Pechenik, Speyer, and Weigandt interpreted \(\nabla\) as a differential operator on Schubert polynomials and used this to prove a new identity for Schubert polynomials and a determinant conjecture of Stanley. In this paper we study a raising operator \(\Delta\) for the strong Bruhat order on the symmetric group, which is in many ways dual to \(\nabla\). We prove a Schubert identity dual to that of \textit{Z. Hamaker} et al. [``Derivatives of Schubert polynomials and proof of a determinant conjecture of Stanley'', Algebr. Comb. (to appear)] and derive formulas for counting weighted paths in the Hasse diagrams of the strong order which agree with path counting formulas for the weak order, providing a strong order analog of Macdonald's reduced word identity. We also show that powers of \(\nabla\) and \(\Delta\) have the same Smith normal forms, which we describe explicitly, answering a question of Stanley. weak order; Bruhat order; Schubert polynomial; duality Combinatorial aspects of representation theory, Classical problems, Schubert calculus A combinatorial duality between the weak and strong Bruhat orders | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Kazhdan-Lusztig polynomials \(P_{x,w}(q)\) play an important role in the study of Schubert varieties as well as the representation theory of semisimple Lie algebras. We give a lower bound for the values \(P_{x,w}(1)\) in terms of ``patterns''. A pattern for an element of a Weyl group is its image under a combinatorially defined map to a subgroup generated by reflections. This generalizes the classical
definition of patterns in symmetric groups. This map corresponds geometrically to restriction to the fixed point set of an action of a one-dimensional torus on the flag variety of a semisimple group \(G\). Our lower bound comes from applying a decomposition theorem for ``hyperbolic localization'' [\textit{T. Braden}, Transform. Groups 8, No. 3, 209-216 (2003; Zbl 1026.14005)] to this torus action. This gives a geometric explanation for the appearance of pattern avoidance in the study of singularities of Schubert varieties. Kazhdan-Lusztig polynomials; Schubert varieties; representations of semisimple Lie algebras; Weyl groups; symmetric groups; flag varieties; semisimple groups; torus actions \beginbarticle \bauthor\binitsS. C. \bsnmBilley and \bauthor\binitsT. \bsnmBraden, \batitleLower bounds for Kazhdan-Lusztig polynomials from patterns, \bjtitleTransform. Groups \bvolume8 (\byear2003), no. \bissue4, page 321-\blpage332. \endbarticle \OrigBibText Sara C. Billey and Tom Braden, Lower bounds for Kazhdan-Lusztig polynomials from patterns , Transform. Groups 8 (2003), no. 4, 321-332. \endOrigBibText \bptokstructpyb \endbibitem Reflection and Coxeter groups (group-theoretic aspects), Grassmannians, Schubert varieties, flag manifolds, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Representation theory for linear algebraic groups Lower bounds for Kazhdan-Lusztig polynomials from patterns. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials For a simple finite dimensional Lie algebra \(\widehat{g}\) of type ADE, let \(g\) be the corresponding (untwisted) affine Lie algebra and \(U_q(\widehat g)\) its quantum affine algebra. In this paper, the author studies finite dimensional representations of \(U_q(\widehat g)\) using geometry of quiver varieties. His purpose is to solve the following conjecture affirmatively, that is, an equivariant \(K\)-homology group of the quiver variety gives the quantum affine algebra \(U_q(\widehat g)\), and to derive results whose analogues are known for \(H_q\).
In \S 1, the author recalls a new realization of \(U_q(\widehat g)\), called Drinfeld realization and introduces the quantum loop algebra \(U_q(Lg)\) as a subquotient of \(U_q(\widehat g)\), which will be studied rather than \(U_q(\widehat g)\). The basic results are recalled on finite dimensional representations of \(U_\varepsilon(Lg)\). And, several useful concepts are introduced.
In \S 2, the author introduces two types of quiver varieties \({\mathcal M}(w)\) and \({\mathcal M}_0(\infty, w)\) as analogues of \(T^*{\mathcal B}\) and the nilpotent cone \(\mathcal N\) respectively. Their elementary properties are given.
In \S 3--\S 8, the author prepares some results on quiver varieties and \(K\)-theory which will be used in later sections.
In \S 9--\S 11, the author considers an analogue of the Steinberg variety
\[
Z(w) = {\mathcal M}(w)\times _{{\mathcal M}_0(\infty,w)}{\mathcal M}(w)
\]
and its equivariant \(K\)-homology \(K^{G_w\times \mathbb{C}^*}(Z(w))\). An algebra homomorphism is constructed from \(U_q(Lg)\) to \(K^{G_w\times \mathbb{C}^*}(Z(W)) \otimes_{\mathbb{Z}[q,q^{-1}]}\mathbb{Q}(q)\).
In \S 12, the author shows that the above homomorphism induces a homomorphism from \(U^\mathbb{Z}_q(Lg)\) to \(K^{G_w\times \mathbb{C}^*}(Z(w))/\text{torsion}\).
In \S 13, the author introduces a standard module \(M_{x,a}\). Thanks to a result in \S 7, it is proved to be isomorphic to \(H_*({\mathcal M}(w)^A_x,\mathbb{C})\) via the Chern character homomorphism. Also, it is shown that \(M_{x,a}\) is a finite dimensional \(l\)-highest weight module. It is conjectured that \(M_{x,a}\) is a tensor product of \(l\)-fundamental representations in some order, which is proved when the parameter is generic in \S 14.1.
In \S 14, it is verified that the standard modules \(M_{x,a}\) and \(M_{y,a}\) are isomorphic if and only if \(x\) and \(y\) are contained in the same stratum. Furthermore, the author shows that the index set \(\{\rho\}\) of the stratum coincides with the set \({\mathcal P} =\{P\}\) of \(l\)-dominant \(l\)-weights of \(M_{0,a}\), the standard module corresponding to the central fiber \(\pi^{-1}(0)\). And, the multiplicity formula \([M(P) : L(Q)] =\dim H^*(i^!_x IC({\mathcal M}^{\text{reg}}_0(\rho_Q)))\) is proved. The result here is simpler than the case of the affine Hecke algebra: nonconstant local systems never appear.
Let \(\text{Res }M(P)\) be the restriction of \(M(P)\) to a \(U_\varepsilon(g)\)-module. In \S\ 15, the author shows the multiplicity formula \([\text{Res }M(P) : L(w - v)] \dim H^*(i_x^! IC({\mathcal M}^{\text{reg}}_0(v, w)))\). This result is compatible with the conjecture that \(M(P)\) is a tensor product of \(l\)-fundamental representations since the restiction of an \(l\)-fundamental representation is simple for type \(A\), and Kostka polynomials give tensor product decompositions.
Two examples are given where \({\mathcal M}^{\text{reg}}_0(v, w)\) can be described explicitly.
As mentioned in the Introduction of this paper, \(U_q(\widehat{g})\) has another realization, called the Drinfeld new realization, which can be applied to any symmetrizable Kac-Moody algebra \(g\), not necessarily a finite dimensional one. This generalization also fits the result in this paper, since quiver varieties can be defined for arbitrary finite graphs. If finite dimensional representations are replaced by \(l\)-integrable representations, parts of the result in this paper can be generalized to a Kac-Moody algebra \(g\), at least when it is symmetric.
If equivariant \(K\)-homology is replaced by equivariant homology, one should get the Yangian \(Y(g)\) instead of \(U_q(\widehat{g})\). The conjecture is motivated again by the analogy of quiver varieties with \(T^*\mathcal B\). As an application, the affirmative solution of the conjecture implies that the representation theory of \(U_q(\widehat g)\) and that of the Yangian are the same. quantum affine algebra; quiver variety; equivariant \(K\)-theory; finite dimensional representation Nakajima, H., Quiver varieties and finite-dimensional representations of quantum affine algebras, \textit{J. Amer. Math. Soc.}, 14, 1, 145-238, (2001) Quantum groups (quantized enveloping algebras) and related deformations, Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory), Group actions on varieties or schemes (quotients), Representations of quivers and partially ordered sets, Connections of basic hypergeometric functions with quantum groups, Chevalley groups, \(p\)-adic groups, Hecke algebras, and related topics Quiver varieties and finite dimensional representations of quantum affine algebras | 0 |
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