pinecone/core-2020-05-10-deduplication huggingface dataset:
New Schur Function Series

AbstractThis paper presents some new product identities for certain summations of Schur functions. These identities are generalizations of some famous identities known to Littlewood and appearing in Macdonald's book. We refer to these identities as the “Littlewood-type formulas.” In addition, analogues for summations of characters of the other classical groups are also given. The Littlewood-type formulas in this paper are separated into two classes, the rational Schur function series and the generalized Schur function series. An application of a rational Schur function series to the infinite product representation of the elliptic theta functions is also given. We prove these Littlewood-type formulas using the Cauchy–Binet formula. The Cauchy–Binet formula is a basic but powerful tool applicable in the present context, which can be derived from our Pfaffian formula, as we explain
Algebraic Criteria to Decide if a Finite Group Acts Effectively on a Model Aspherical Manifold

AbstractFor an aspherical manifoldM, arising from a Seifert fiber space construction, it is known that, under some additional conditions onM, a finite abstract kernel ψ:F→Out(π1(M)) can be (effectively) geometrically realized by a group of fiber preserving homeomorphisms ofMif and only if ψ can be realized by an (admissible) group extension 1→π1(M)→E→F→1. Hence, the study of the symmetry of such a manifold (in terms of finite effective actions onM) can be converted into a group-theoretical study of realizing (algebraically) finite abstract kernelsF→Out(π1(M)). This question, conceptually, is well understood: there exists an extension realizing a given abstract kernel if and only if the corresponding third cohomology class, called the obstruction, vanishes. Unfortunately, a straightforward computation of this obstruction can be extremely hard. In this paper, we present some criteria which solve this problem for certain finite abstract kernels. Instead of using cohomological arguments, a completely different, rather technical and computational approach, based on the Reidemeister–Schreier method for presenting subgroups of finite index in a given finitely presented group, is followed. Not only the actual results but also this approach is of interest since it certainly allows one to produce similar criteria for other finite groups
Two-Sided Tilting Complexes for Gorenstein Orders

AbstractWork of J. Rickard proves that the derived module categories of two ringsAandBare equivalent as triangulated categories if and only if there is a particular objectT, a so-called tilting complex, in the derived category ofAsuch thatBis the endomorphism ring ofT. The functor inducing the equivalence, however, is not explicit by the knowledge ofT. Suppose the derived categories ofAandBare equivalent. IfAandBareR-algebras and projective of finite type over the commutative ringR, then Rickard proves the existence of a so-called two-sided tilting complexX, which is an object in the derived category of bimodules. The left derived tensor product byXis then an equivalence between the desired categories ofAandB. There is no general explicit construction known to deriveXfrom the knowledge ofT. In an earlier paper S. König and the author gave for a class of algebras a tilting complexTby a general procedure with prescribed endomorphism ring. Under some mild additional hypotheses, we construct in the present paper an explicit two-sided tilting complex whose restriction to one side is any given one-sided tilting complex of the type described in the above-cited work. This provides two-sided tilting complexes for various cases of derived equivalences, making the functor inducing this equivalence explicit. In particular, the perfect isometry induced by such a derived equivalence is determined
Simple Lie Color Algebras of Witt Type

AbstractLetKbe a field and let ε: Γ×Γ→K•be a bicharacter defined on the multiplicative group Γ. We suppose thatAis a Γ-graded, associativeK-algebra that is color commutative with respect to ε. Furthermore, let Δ be a nonzero Γ-graded,K-vector space of color derivations ofAand suppose that Δ is also color commutative with respect to the bicharacter ε. Then, with a rather natural definition,A⊗KΔ=AΔ becomes a Lie color algebra, and we obtain necessary and sufficient conditions here for this Lie color algebra to be simple. With two minor exceptions when dimKΔ=1, simplicity occurs if and only ifAis graded Δ-simple andAΔ⊗Δ=AΔΔ acts faithfully as color derivations onA
Identities for Classical Group Characters of Nearly Rectangular Shape

AbstractWe derive several identities that feature irreducible characters of the general linear, the symplectic, the orthogonal, and the special orthogonal groups. All of the identities feature characters that are indexed by shapes that are “nearly” rectangular, by which we mean that the shapes are rectangles except for one row or column that might be shorter than the others. As applications we prove new results in plane partitions and tableaux enumeration, including new refinements of the Bender–Knuth and MacMahon (ex-)conjectures
The Infinitude of 7-Arc-Transitive Graphs

AbstractBy theorems of Tutte, Weiss, and others, it is known that there are no finite symmetric graphs of degree greater than 2 with automorphism group transitive on 8-arcs, and that 7-arc-transitivity can occur only in the case of graphs of degree 3m+1. In this article it is shown that there are infinitely many 7-arc-transitive finite quartic graphs; indeed for all but finitely many positive integersn, there is a finite connected 7-arc-transitive quartic graph with the alternating groupAnacting transitively on its 7-arcs, and another with the symmetric groupSnacting transitively on its 7-arcs. The proof uses a construction involving permutation representations of a generic infinite group to produce an infinite family of finite graphs with the required properties
The Transitivity of Local Warfield Groups

AbstractFor a fixed primep, letGdenote a module over the integers localized atp. Such a module is often referred to as ap-local (abelian) group. Following established precedent, we say thatGis transitive provided that there exists an automorphism ofGthat mapsaontobwheneveraandbare elements ofGhaving the same height sequence. However, relatively few nontorsionGmeet this criterion of transitivity. Indeed, as we show, there are four different classes of simply presentedp-local groups of torsion-free rank 3 that are nontransitive. In this article, we introduce a weaker version of transitivity that allows all local Warfield groups (=summands of simply presentedp-local groups) to qualify as being transitive. In this connection, we associate with each element of a local Warfield groupGa new invariant, called a type vector, and we prove that there is an automorphism ofGthat mapsaontobif and only ifaandbhave the same height sequence and the same type vector. As an application of this result, we are able to determine exactly which local Warfield groups are transitive in the classical sense. Also, our results yield an alternate proof of a recent result of Files regarding the equivalence of transitivity and full transitivity for local Warfield groups. Finally, we give a complete survey of transitivity for local Warfield groups of torsion-free rank 3
On Quotients of Hom-Functors and Representations of Finite General Linear Groups II

AbstractThis is a second article on quotients of Hom-functors and their applications to the representation theory of finite general linear groups in a nondescribing characteristic. After some general results on quotients of Hom-functors and their connection to the Harish–Chandra theory these constructions are used to obtain a full classification of the ℓ-modular irreducible representations ofGLn(q) for some prime powerqwhich is not divisible by the prime ℓ and to explain some facts on their Harish–Chandra series and decomposition numbers
Derived Equivalence Classification of Algebras of Dihedral, Semidihedral, and Quaternion Type

AbstractAlgebras of dihedral, semidihedral, and quaternion type were introduced by K. Erdmann as a generalization of blocks of finite groups of tame representation type. In a series of papers Erdmann gave a description of the Morita equivalence classes of these algebras. The aim of the present article is to present a classification of algebras of dihedral, semidihedral, and quaternion type up to derived equivalence. As an application we can show that all basic algebras in Erdmann's list are actually of tame representation type
Some Results About the Quillen Complex ofSp2n(q)

AbstractThe structure of the Quillen complex ofSp2n(q) atp, denoted Ap(Sp2n(q)), is known whenpis the characteristic prime. In this paper it is shown that ifp∣(q−1) then Ap(Sp2n(q)) is Cohen–Macaulay of dimension (n−1). Furthermore, ifd≥3 is the order ofqin Z/pZ anddis odd, it is shown that Ap(Sp2n(q)) is simply connected whenevermp(Sp2n(q))≥3.It is also shown that the order complex of proper, nondegenerate subspaces of a 2n-dimensional symplectic space over Fq—ordered by inclusion—is Cohen–Macaulay of dimension (n−2) ifq≠2
Semisimple Commutative Algebras with Positive Bases

AbstractAlgebras that serve as models for concurrent studying of certain aspects of both the algebra of ordinary characters and the center of the group algebra of a finite group have been considered by various authors. In this article we offer another such model. The main differences between our model and the known ones are: 1. Our model includes Brauer characters and principal indecomposable characters as special cases. 2. Our emphasis is on the eigenvalues of the regular representation of the algebra elements, an approach that gives results on values of characters (ordinary, central, Brauer, principal indecomposable) as well as on their products
Primitive Algebras with Arbitrary Gelfand-Kirillov Dimension

AbstractWe construct, for every real β ≥ 2, a primitive affine algebra with Gelfand-Kirillov dimension β. Unlike earlier constructions, there are no assumptions on the base field. In particular, this is the first construction over ℝ or L. Given a recursive sequence {vn} of elements in a free monoid, we investigate the quotient of the free associative algebra by the ideal generated by all nonsubwords in {vn}. We bound the dimension of the resulting algebra in terms of the growth of {vn}. In particular, if ⋎νn⋎ is less than doubly exponential, then the dimension is 2. This also answers affirmatively a conjecture of Salwa (1997, Comm. Algebra 25, 3965–3972)
Structurable algebras and groups of type E_6 and E_7

It is well-known that every algebraic group of type F_4 is the automorphism
group of an exceptional Jordan algebra, and that up to isogeny all groups of
type ^1E_6 with trivial Tits algebras arise as the isometry groups of norm
forms of such Jordan algebras. We describe a similar relationship between
groups of type E_6 and groups of type E_7 and use it to give explicit
descriptions of the homogeneous projective varieties associated to groups of
type E_7 with trivial Tits algebras. The underlying algebraic structure for the
relationship considered here are a sort of 56-dimensional structurable algebra
which are forms of an algebra constructed from an exceptional Jordan algebra.Comment: 35 pages, AMSLaTeX -- error in final section correcte
On fields of moduli of curves

AbstractThe field of moduli K of a curve X a priori defined over the separable closure K5 of K need not be a field of definition. This paper shows that the obstruction is essentially the same as the obstruction to K being a field of definition of the cover X → X/Aut(X). Using previous results of Dèbes-Douai, we then obtain a cohomological measure of the obstruction. This yields concrete criteria for the field of moduli to be a field of definition. An interesting application is the following local-global principle. If a curve X, together with all of its automorphisms, is defined over ℚp for all primes p, then it is defined over ℚ
Minimal Resolutions of Algebras

AbstractA method is described for constructing the minimal projective resolution of an algebra considered as a bimodule over itself. The method applies to an algebra presented as the quotient of a tensor algebra over a separable algebra by an ideal of relations that is either homogeneous or admissible (with some additional finiteness restrictions in the latter case). In particular, it applies to any finite-dimensional algebra over an algebraically closed field. The method is illustrated by a number of examples, viz. truncated algebras, monomial algebras, and Koszul algebras, with the aim of unifying existing treatments of these in the literature
Hopf algebras of low dimension

The main aim of this paper is to classify all types of Hopf algebras of dimension less thn or equal to 11 over an algebraically closed field of characteristic 0. If A is such a Hopf algebra that is not semisimple, then we shall prove that A or A* is pointed. This property will result from the fact that, under some assumptions, any Hopf algebra that is generated as an algebra by a four-dimensional simple subcoalgebra is a Hopf quotient of the coordinate ring of quantum SL2(k). The first result allows us to reduce the classification to the case of pointed Hopf algebras of dimension 8. We shall describe their types in the last part of the paper
On theq-Commutations inUq(n) at Roots of One

AbstractLet ε be a root of one and g a semisimple Lie algebra with triangular decomposition g=n+h+n−. LetU+ε(resp.Ures+ε) be the nonrestricted (resp. restricted) quantum enveloping algebra of n. We prove that FractU+εis a quantum Weyl field. We then give a description of the ε-center ofU+ε. LetUfin+εbe the finite part ofUres+ε. Via the Drinfeld correspondence, theUfin+ε-covariant space of a Weyl module is ε-central. In case g=sln, this enables us to describe this space in terms of semistandard Young tableaux
TKK algebras and vertex operator representations

Let  be the Jordan algebra constructed from a semilattice S of ℝν (v ≥ 1) [see Allison et al., Mem. Amer. Math. Soc., 603 (1997)]. Let  be the Lie algebra obtained from the Jordan algebra  by the Tits-Kantor-Koecher construction. The TKK algebra  is defined to be the universal central extension of the Lie algebra . In this paper we present a complete description of the TKK algebra , where S is the smallest possible (nonlattice) semilattice, which then allows us to give a representation to this Lie algebra by vertex operators
Transcendence of Binomial and Lucas' Formal Power Series

AbstractThe formal power series[formula]is transcendental over Q(X) whentis an integer ≥2. This is due to Stanley forteven, and independently to Flajolet and to Woodcock and Sharif for the general case. While Stanley and Flajolet used analytic methods and studied the asymptotics of the coefficients of this series, Woodcock and Sharif gave a purely algebraic proof. Their basic idea is to reduce this series modulo prime numbersp, and to use thep-Lucas property: ifn=∑nipiis the basepexpansion of the integern, then[equation]The series reduced modulopis then proved algebraic over Fp(X), the field of rational functions over the Galois field Fp, but its degree is not a bounded function ofp. We generalize this method to characterize all formal power series that have thep-Lucas property for “many” prime numbersp, and that are furthermore algebraic over Q(X)
Some model theory over hereditary noetherian domains

Questions in the model theory of modules over hereditary noetherian domains are investigated with particular attention being paid to differential polynomial rings and to generalized Weyl algebras. We prove that there exists no isolated point in the Ziegler spectrum over a simple hereditary generalized Weyl algebra A of the sort considered by Bavula [Algebra i Analiz 4(1) (1992), 75–97] over a field k with char(k) = 0 (the first Weyl algebra A1(k) is one such) and the category of finite length modules over A does not have any almost split sequence. We show that the theory of all modules over a wide class of generalized Weyl algebras and related rings interprets the word problem for groups, and in the case that the field is countable there exists a superdecomposable pure-injective module over A. This class includes, for example, the universal enveloping algebra Usl2(k)
The Action ofF4(q) on Cosets ofB4(q)

AbstractIn this paper we consider the action of the simple groupF4(q) on the cosets of the maximal subgroupB4(q). We show that the action is multiplicity-free of rankq+3; we obtain suborbit representatives and calculate subdegrees, show that all suborbits are self-paired, find that none of the graphs arising from the action is distance-transitive, and give explicitly the decomposition of the permutation character. In addition, we give detailed information on the correspondence between geometric conjugacy classes and semisimple classes which is used in the Deligne–Lusztig theory
Hecke Algebras of TypeDnat Roots of Unity

AbstractThis paper deals with representations of Hecke algebras of typeDn(i.e., Hq(Dn)) over any fieldKand with any parameterq. Our method is to study the restrictions (to Hq(Dn)) of the Specht moduleS̃λand simple moduleD̃λof Hq,1(Bn). We prove thatS̃λ|Hq(Dn)≅S̃λ̂|Hq(Dn), whereλ̂=(λ(2),λ(1)). When λ(1)=λ(2), we show thatS̃λis a direct sum of its two Hq(Dn)-submodules with the same dimensions. Via this we give a crude presentations of all the simple modules of Hq(Dn). When Hq(Dn) is semisimple, we give a complete set of pairwise non-isomorphic simple modules. Finally, we also give an attempt of using “Murphy basis” philosophy and construct a basis for Hq(Dn) whennis odd
Invariant Algebra and Cuspidal Representations of Finite Monoids

AbstractMotivated by the theories of Hecke algebras and Schur algebras, we consider in this paper the algebra CMGofG-invariants of a finite monoidMwith unit groupG. IfMis a regular “balanced” monoid, we show that CMGis a quasi-hereditary algebra. In such a case, we find the blocks of CMGto be the “sections” of the blocks of CM. We go on to develop a theory of cuspidal representations for balanced monoids. We then apply our results to the full transformation semigroup and the multiplicative monoid of triangular matrices over a finite field
Lifting of Quantum Linear Spaces and Pointed Hopf Algebras of Orderp3

AbstractWe propose the following principle to study pointed Hopf algebras, or more generally, Hopf algebras whose coradical is a Hopf subalgebra. Given such a Hopf algebraA, consider its coradical filtration and the associated graded coalgebra grA. Then grAis a graded Hopf algebra, since the coradicalA0ofAis a Hopf subalgebra. In addition, there is a projection π: grA→A0; letRbe the algebra of coinvariants of π. Then, by a result of Radford and Majid,Ris a braided Hopf algebra and grAis the bosonization (or biproduct) ofRandA0: grA≃R#A0. The principle we propose to studyAis first to studyR, then to transfer the information to grAvia bosonization, and finally to lift toA. In this article, we apply this principle to the situation whenRis the simplest braided Hopf algebra: a quantum linear space. As consequences of our technique, we obtain the classification of pointed Hopf algebras of orderp3(pan odd prime) over an algebraically closed field of characteristic zero; with the same hypothesis, the characterization of the pointed Hopf algebras whose coradical is abelian and has indexporp2; and an infinite family of pointed, nonisomorphic, Hopf algebras of the same dimension. This last result gives a negative answer to a conjecture of I. Kaplansky
Minuscule Elements of Weyl Groups, the Numbers Game, andd-Complete Posets

AbstractCertain posets associated to a restricted version of the numbers game of Mozes are shown to be distributive lattices. The posets of join irreducibles of these distributive lattices are characterized by a collection of local structural properties, which form the definition ofd-complete poset. In representation theoretic language, the top “minuscule portions” of weight diagrams for integrable representations of simply laced Kac–Moody algebras are shown to be distributive lattices. These lattices form a certain family of intervals of weak Bruhat orders. These Bruhat lattices are useful in studying reduced decompositions of λ-minuscule elements of Weyl groups and their associated Schubert varieties. Thed-complete posets have recently been proven to possess both the hook length and the jeu de taquin properties
Rings with Auslander Dualizing Complexes

AbstractA ring with an Auslander dualizing complex is a generalization of an Auslander–Gorenstein ring. We show that many results which hold for Auslander–Gorenstein rings also hold in the more general setting. On the other hand we give criteria for existence of Auslander dualizing complexes which show these occur quite frequently. The most powerful tool we use is the Local Duality Theorem for connected graded algebras over a field. Filtrations allow the transfer of results to nongraded algebras. We also prove some results of a categorical nature, most notably the functoriality of rigid dualizing complexes
A Probabilistic Approach Toward Conjugacy Classes in the Finite General Linear and Unitary Groups

AbstractThe conjugacy classes of the finite general linear and unitary groups are used to define probability measures on the set of all partitions of all natural numbers. Probabilistic algorithms for growing random partitions according to these measures are obtained. These algorithms are applied to prove group theoretic results which are typically proved by techniques such as character theory and Moebius inversion. Among the theorems studied are Steinberg's count of unipotent elements, Rudvalis' and Shinoda's work on the fixed space of a random matrix, and Lusztig's count of nilpotent matrices of a given rank. Generalizations of these algorithms based on Macdonald's symmetric functions are given
Cuspidal Representations Associated to (GL(n),O(n)) over Finite Fields andp-Adic Fields

AbstractWhenr̄is an irreducible cuspical representation ofḠ=GL(n,q) andH̄is an orthogonal group associated to a symmetric matrix inḠthen the space ofH̄-fixed vectors forr̄is shown to have dimension at most one. Such a representationr̄induces an irreducible supercuspidal representation π ofG=GL(n,E), whereEis ap-adic field whose residue field has orderq. The space of those linear forms on the space of π which are invariant under an orthogonal group is computed. For the corresponding group of orthogonal similitudes, it is shown that the dimension of the space of invariant linear forms is always at most one
Extended Central Characters and Dixmier's Map

AbstractLet p be a parabolic subalgebra in a semisimple Lie algebra g. We study the moduleMp(λ) induced from a one-dimensional p-module of weight λ. LetU=U(g)/Ibe the quotient of the enveloping algebraU(g) by the annihilatorIof the generic module induced from p. LetZ̃denote the integral closure of the centerZofU, andŨ=Z̃U. ThenMp(λ) is not only aU-module, but even aŨ-module [1], so its central characterZ→C extends to a central characterZ̃→C. We prove that this extended central character is useful: It is unique if and only if Dixmier's map is injective. Moreover, the “modified Dixmier-map” of the author's work (1998,Abh. Math. Sem. Univ. Hamburg68, 25–44) from the sheetS/Gdetermined by p into the space XŨof minimal primitive ideals ofŨis a homeomorphismS/G→XŨgiven by O↦AnnMp(λ). As an application, we obtain the following result related to Duflo's theorem that minimal primitive ideals are induced and centrally generated: Each minimal primitive idealJofŨis induced and “almost” generated by its intersection m with the center, in the sense thatJ=mŨ. As further application of the main theorem we get results on the well-definedness of Dixmier's map