pinecone/core-2020-05-10-deduplication huggingface dataset:
Schur Subalgebras and an Application to the Symmetric Group

AbstractLet K be an infinite field of prime characteristic p and let d≤r be positive integers of the same parity satisfying a certain congruence condition. We prove that the Schur algebra S(2,d) is isomorphic to a subalgebra of the form eS(2,r)e, where e is a certain idempotent of S(2,r). Translating this result via Ringel duality to the symmetric groups Σd and Σr, we obtain lattice isomorphisms between Specht modules, between Young modules, and between permutation modules. Here modules labelled by the partitions (r−k,k) correspond to modules labelled by (d−k,k). This provides a representation theoretical interpretation for part of the fractal structures observed for the decomposition numbers of the symmetric groups corresponding to two-part partitions
General Formulas for Solving Solvable Sextic Equations

AbstractLet G be a transitive, solvable subgroup of S6. We show that there is a common formula for finding the roots of all irreducible sextic polynomials f(x)∈Q[x] with Gal(f)=G. Moreover, once the roots ri are calculated, there is an explicit procedure for numbering them so that the Galois group acts via τ(ri)=rτ(i) for τ∈G⊂S6. We also demonstrate new criteria for determining the Galois group of an irreducible sextic polynomial f(x)∈Q[x]. The first two results generalize results of D. S. Dummit (1991, Math. Comp.57, 387–401) for quintic polynomials
Conjugacy Classes in Maximal Parabolic Subgroups of General Linear Groups

AbstractWe compute conjugacy classes in maximal parabolic subgroups of the general linear group. This computation proceeds by reducing to a “matrix problem.” Such problems involve finding normal forms for matrices under a specified set of row and column operations. We solve the relevant matrix problem in small dimensional cases. This gives us all conjugacy classes in maximal parabolic subgroups over a perfect field when one of the two blocks has dimension less than 6. In particular, this includes every maximal parabolic subgroup of GLn(k) for n<12 and k a perfect field. If our field is finite of size q, we also show that the number of conjugacy classes, and so the number of characters, of these groups is a polynomial in q with integral coefficients
On the Variety Determined by Symmetric Quadratic Algebras

AbstractWe consider some polynomial identities of degree ≤5 which are satisfied by all symmetric quadratic algebras. We call rings satisfying these identities generalized quadratic rings, or GQ-rings. We show that when the ring is not flexible, these identities are enough to make the ring quadratic over its center. Therefore, simple nonflexible GQ-rings are symmetric quadratic algebras over their center, which is a field. For prime GQ-rings, the center has no nonzero zero divisors. Prime GQ-rings, which are not flexible, are subrings of the quadratic algebra formed by extending the center to its field of quotients. Flexible GQ-rings are noncommutative Jordan rings which satisfy a quadratic condition over their commutative center. We show that any semi-prime GQ-ring is a subdirect sum of a noncommutative Jordan ring (which satisfies a quadratic condition over its commutative center) and a nonflexible ring (which is symmetric quadratic over its center)
Derivation-Simple Algebras and the Structures of Lie Algebras of Witt Type

AbstractWe classify all the pairs of a commutative associative algebra with an identity element and its finite-dimensional locally finite Abelian derivation subalgebra such that the commutative associative algebra is derivation-simple with respect to the derivation subalgebra over an algebraically closed field with characteristic 0. Such pairs are the fundamental ingredients for constructing simple Lie algebras of Cartan type. Moreover, we determine the isomorphism classes of the simple Lie algebras of Witt type. The structure space of these algebras is given explicitly
Noncommutative Symmetric Algebras of Two-Sided Vector Spaces

AbstractWe generalize the definition of the symmetric algebra of a vector space in order to define noncommutative symmetric algebras of two-sided vector spaces. Not all two-sided vector spaces have noncommutative symmetric algebras; the ones that do are called admissible, and conditions for admissibility are given. Further, for some classes of admissible two-sided vector spaces, the skew fields of fractions of their noncommutative symmetric algebras are computed. The degree 0 components of these skew fields correspond to function fields of certain noncommutative ruled surfaces, and hence allow us to determine birational equivalence classes for such surfaces
Explicit Constructions of the Fundamental Representations of the Symplectic Lie Algebras

AbstractWe give two constructions for each fundamental representation of sp(2n,C). We also present quantum versions of these constructions. These are explicit in the sense of the Gelfand–Tsetlin constructions of the irreducible representations of gl(n,C): we explicitly specify the matrix elements for certain generators of sp(2n,C) with respect to each of the two explicit bases presented. In fact, our constructions appear to have been the first such infinite family of explicit constructions of irreducible representations of simple Lie algebras since the Gelfand–Tsetlin constructions were obtained in 1950. Our approach is combinatorial; the key idea is to find a suitable family of partially ordered sets on which to present the action of the Lie algebra, and then to use these posets to produce the bases and the actions of the generators. Our constructions of the fundamental representations of sp(2n,C) take place on two families of posets which we call “symplectic lattices.” Previously (1999, J. Combin. Theory Ser. A88, 217–234), we used these representation constructions to confirm a conjecture of Reiner and Stanton concerning one of these families of lattices
Elimination of Variables in Linear Solvable Polynomial Algebras and ∂-Holonomicity

AbstractLet k be a field of characteristic 0. Based on the Gelfand–Kirillov dimension computation of modules over solvable polynomial k-algebras, where solvable polynomial algebras are in the sense of A. Kandri-Rody and V. Weispfenning (1990, J. Symbolic Comput.9, 1–26), we prove that the elimination lemma, obtained from D. Zeilberger (1990, J. Comput. Appl. Math.32, 321–368) by using holonomic modules over the Weyl algebra An(k) and used in the automatic proving of special function identities, holds for a class of solvable polynomial k-algebras without any “holonomicity” restriction. This opens a way to the solution of the extension/contraction problem stemming from the automatic proving of multivariate identities with respect to the ∂-finiteness in the sense of F. Chyzak and B. Salvy (1998, J. Symbolic Comput.26, 187–227). It also yields a ∂-holonomicity so that automatic proving of multivariate identities may be dealt with by manipulating polynomial function coefficients instead of rational functions
Determination of All Q-Rational CM-Points in the Moduli Space of Principally Polarized Abelian Surfaces

AbstractIt is known that in the moduli space A1 of elliptic curves, there exist precisely 9 Q-rational points corresponding to the isomorphism class of elliptic curves with complex multiplication by the ring of algebraic integers of a principal imaginary quadratic number field. Here, we prove that in the moduli space A2 of principally polarized abelian surfaces, there exist precisely 19 Q-rational points corresponding to the isomorphism class of abelian surfaces whose endomorphism rings are isomorphic to the rings of algebraic integers of some imaginary cyclic quartic number fields
On the Goldie Quotient Ring of the Enveloping Algebra of a Classical Simple Lie Superalgebra

AbstractIf g is a classical simple Lie superalgebra (g≠P(n)), the enveloping algebra U(g) is a prime ring and hence has a simple artinian ring of quotients Q(U(g)) by Goldie's Theorem. We show that if g has Type I then Q(U(g)) is a matrix ring over Q(U(g0)). On the other hand, if g=osp(1,2r) then by extending the center of U(g) we obtain a prime ring whose Goldie quotient ring is a matrix ring over the quotient division ring of a Weyl algebra. This is an analog of a result of Gelfand and Kirillov
The Cohomology Structure of an Algebra Entwined with a Coalgebra

AbstractTwo cochain complexes are constructed for an algebra A and a coalgebra C entwined with each other via the map ψ:C⊗A→A⊗C. One complex is associated to an A-bimodule, the other to a C-bicomodule. In the former case the resulting complex can be considered as a ψ-twisted Hochschild complex of A, while for the latter one obtains a ψ-twist of the Cartier complex of C. The notion of a weak comp algebra is introduced by weakening the axioms of the Gerstenhaber comp algebra. It is shown that such a weak comp algebra is a cochain complex with two cup products that descend to the cohomology. It is also shown that the complexes associated to an entwining structure and A or C are examples of a weak comp algebra. Finally both complexes are combined in a double complex whose role in the deformation theory of entwining structures is outlined
Realization of the Space of Conformal Blocks in Lie Algebra Modules

AbstractUsing integrable irreducible representations of generalized twisted affine Lie algebra modules, we give a realization of the space of conformal blocks of conformal field theory on a stable algebraic curve. Many basic properties of the conformal blocks, such as finite dimensionality of the space, invariance of the conformal blocks under suitable formal neighborhood changes, and the property of “propagation of vacua” are discussed. Finally, a relative local 1-form around a fixed point of the order two automorphism of the curve is given
Field Theory and the Cohomology of Some Galois Groups

AbstractWe prove that two arithmetically significant extensions of a field F coincide if and only if the Witt ring WF is a group ring Z/n[G]. Furthermore, working modulo squares with Galois groups which are 2-groups, we establish a theorem analogous to Hilbert's Theorem 90 and show that an identity linking the cohomological dimension of the Galois group of the quadratic closure of F, the length of a filtration on a certain module over a Galois group, and the dimension over F2 of the square class group of the field holds for a number of interesting families of fields. Finally, we discuss the cohomology of a particular Galois group in a topological context
On Klein Surfaces with 2p or 2p+2 Automorphisms

AbstractIt is known that the maximal order of a cyclic group of automorphisms admitted by a bordered Klein surface or real algebraic curve of algebraic genus p is 2p or 2(p+1), depending on whether p is odd or even. In this paper, we classify the automorphism groups of all bordered Klein surfaces or real algebraic curves which admit an automorphism group of order 2p, if p is odd, or 2(p+1), if p is even. The topological type of each of these surfaces is determined. The defining equations of these real algebraic curves are presented in all but two cases. For these cases examples are given
Primeness Criteria for Universal Enveloping Algebras of Lie Color Algebras

AbstractLet L=L+⊕L− be a Lie color algebra with dimL−<∞. We write detL≠0 if the matrix formed by brackets between elements of a basis of L− is nonsingular. Unlike Lie super algebras, a Lie color algebra L may have detL≠0 and a universal enveloping algebra U(L) which is not prime. We will provide examples and show that U(L) is semiprime whenever detL≠0. Our main theorem is a criterion for U(L) to be prime. As a corollary, we prove that U(L) is prime whenever detL≠0 and the grading group G is either a finite group whose 2-torsion subgroup is cyclic or a finitely generated group such that for each elementary divisor 2l of G the base field does not contain a primitive (2l)th-root of unity
Action of Hecke Correspondences on Heegner Curves on a Siegel Threefold

AbstractWe get an analog of Kolyvagin's trace relations for a Siegel threefold X. Namely, let V⊂X be a Heegner curve (points of V correspond to Abelian surfaces with some fixed multiplication ring) and let Tp be a Hecke correspondence on X, so Tp(V) is a codimension 2 cycle on X. We describe the set of irreducible components of the support of Tp(V) in terms of geometry of Tp(t), where t∈X is a generic point. Tp(t) is a three-dimensional quadric hypersurface over Fp. We find also some equivalence relations on this set of irreducible components.The same method can be applied to other pairs V⊂X (or more generally chains Xn⊂Xn−1⊂···⊂X1) of Shimura varieties, and other Hecke correspondences. Finally, we discuss the possibility of finding reductions at p of these irreducible components, and of applying the Birch–Mazur–Bloch method to prove that the Abel–Jacobi image of some linear combination of V and other similar curves is not of torsion
Triangular Factorizations of Special Polynomial Automorphisms

AbstractIn this paper we give explicit factorizations which demonstrate the stable tameness of all polynomial automorphisms arising from a recent construction of Hubbers and van den Essen. This is accomplished by two different factorizations of such an automorphism by triangular automorphisms, one which is concise but requires a large number of additional dimensions, and one which is more intricate but requires fewer dimensions. The stable tameness of automorphisms in this special class aligns with other existing evidence to suggest that all polynomial automorphisms may be stably tame
A Straightening Algorithm for Row-Convex Tableaux

AbstractWe produce a new basis for the Schur and Weyl modules associated to a row-convex shape D. The basis is indexed by a new class of “straight” tableaux which we introduce by weakening the usual requirements for standard tableaux. Spanning is proved via a new straightening algorithm for expanding elements of the representation into this basis. For skew shapes, this algorithm specializes to the classical straightening law. The new straight basis is used to produce bases for flagged Schur and Weyl modules, to provide Groebner and SAGBI bases for the homogeneous coordinate rings of some configuration varieties, and to produce a flagged branching rule for row-convex representations. Systematic use of supersymmetric letterplace techniques enables the representation theoretic results to be applied to representations of the general linear Lie superalgebra as well as to the general linear group
Relatively Free Semigroups of Intermediate Growth

AbstractThe problem of calculating the growth of a finitely generated (f.g.) semigroup satisfying the given system of identities is considered. Examples of relatively free semigroups having intermediate growth, new growth criteria, and constructions are given. The main ideas employed in the proofs are based on some new points of view on growth. Our method is also useful for constructing various examples of f.g. nilpotent and nil-semigroups having intermediate growth and arbitrarily small Gelfand–Kirillov superdimension
Half-Factorial Domains in Quadratic Fields

AbstractIn this paper, we give a norm-theoretic proof of the characterization of HFDs that appear as orders in quadratic fields. The original characterization of orders that are HFDs in quadratic fields is due to F. Halter-Koch (1983, Ber. Mat. Stat. Sektion Forschung.191). Our approach will pay special attention to the normset of such HFDs, and we will give a characterization that relates to the generalized class number of such an order. Additionally, it will be shown that such HFDs are characterized by the factorization properties of their normsets
New Invariants of Noetherian Local Rings

AbstractWe investigate properties of certain invariants of Noetherian local rings, including their behavior under flat local homomorphisms. We show that these invariants are bounded by the multiplicity for Cohen–Macaulay local rings with infinite residue fields, and they all agree with the multiplicity when such rings are hypersurfaces. We also show that these invariants are all equal to 2 for a non-regular Cohen–Macaulay local ring A if and only if A has a minimal multiplicity, provided its residue field is infinite
Test Modules to Calculate Dutta Multiplicities

AbstractUsing Frobenius maps, the Dutta multiplicity χ∞(F) was first defined by Dutta for a complex F over a local ring of positive characteristic. Nowadays, using localized Chern characters or Adams operations, it is defined for a complex not only over a local ring of positive characteristic but also over a ring that is a homomorphic image of an arbitrary regular local ring. In the paper, we shall give another characterization to Dutta multiplicity using some Galois extension.Furthermore, we shall define the notion of test modules. If a test module for a local ring exists, then the positivity conjecture of Dutta multiplicities is true over the ring. In the paper, it is proved that, if the small Cohen–Macaulay modules conjecture is true, then test modules always exist for complete local domains whose field of fractions are of characteristic 0
Lie Algebras Generated by Extremal Elements

AbstractWe study Lie algebras generated by extremal elements (i.e., elements spanning inner ideals) over a field of characteristic distinct from 2. There is an associative bilinear form on such a Lie algebra; we study its connections with the Killing form. Any Lie algebra generated by a finite number of extremal elements is finite dimensional. The minimal numbers of extremal generators for the Lie algebras of type An (n≥1), Bn (n≥3), Cn (n≥2), Dn (n≥4), En (n=6,7,8), F4 and G2 are shown to be n+1, n+1, 2n, n, 5, 5, and 4 in the respective cases. These results are related to group theoretic ones for the corresponding Chevalley groups
Low-Dimensional Representations of Special Unitary Groups

AbstractWe prove gap results for the low-dimensional representation of unitary groups in nondefining characteristics. More precisely, we give a lower bound for the third smallest degree of a nontrivial absolutely irreducible representation of the groups mentioned above, which is almost in the order of magnitude of the square of the two smallest such representations. As a corollary we obtain the uniqueness of Weil representations in all cross characteristics for the unitary groups. Our approach uses some results on ordinary representations which can be proved using Deligne–Lusztig theory and which might be of independent interest, as well as information on decomposition matrices
On Bounding the Number of Generators for Fat Point Ideals on the Projective Plane

AbstractLet X be the surface obtained by blowing up general points p1,…,pn of the projective plane over an algebraically closed ground field k, and let L be the pullback to X of a line on the plane. If C is a rational curve on X with C·L=d, then for every t there is a natural map Γ(C,OC(t))⊗Γ(X,OX(L))→Γ(C,OC(t+d)) given by multiplication on simple tensors. The ranks of such maps are determined as a function of t, d, and m, where m is the largest multiplicity of C at any of the points pi. If I is the ideal defining the fat point subscheme Z=m1p1+···+mnpn⊂P2, and α is the least degree in which I has generators, then the ranks of the maps Γ(C,OC(t))⊗Γ(X,OX(L))→Γ(C,OC(t+d)) can be used for bounding the number of generators of I in degrees t>α+1
Markov Traces and Generic Degrees in Type Bn

AbstractMotivated by the theory of knots, Geck and Lambropoulou studied so-called Markov traces on the Iwahori–Hecke algebra Hn of type Bn. These traces depend on two parameters and are linear combinations of the irreducible characters of Hn, where the coefficients are called weights. Orellana determined the weights explicitly for certain special choices of the parameters. In this article, we derive from Orellana's result a formula for the weights in general. As an application, we obtain a new proof of Hoefsmit's formulas for the generic degrees of Hn. Finally, we present a conjectural formula for weights of Markov traces on Ariki–Koike algebras
Some Simple Projective Brauer Quotients of Simple Modules for the Symmetric Groups in Characteristic Two

AbstractBy Alperin's weight conjecture (J. L. Alperin and P. Fong, 1990, J. Algebra131, 2–22) the number of simple kSn-modules equals the number of weights for Sn, where Sn is the symmetric group on n symbols and k is a field of characteristic p>0. In this paper we answer the question, “When is the Brauer quotient of a simple F2Sn-module V with respect to a subgroup H of Sn both simple and projective as an NSn(H)/H-module?,” in some special cases. Remarkably, in each case there is only one such subgroup H (up to conjugacy)
Regions of Linearity, Lusztig Cones, and Canonical Basis Elements for the Quantized Enveloping Algebra of Type A4

AbstractLet Uq be the quantum group associated to a Lie algebra g of rank n. The negative part U− of U has a canonical basis B with favourable properties (see M. Kashiwara (1991, Duke Math. J.63, 465–516) and G. Lusztig (1993. “Introduction to Quantum Groups,” Sect. 14.4.6, Birkhäuser, Boston)). The approaches of Lusztig and Kashiwara lead to a set of alternative parametrizations of the canonical basis, one for each reduced expression for the longest word in the Weyl group of g. We show that if g is of type A4 there are close relationships between the Lusztig cones, canonical basis elements, and the regions of linearity of reparametrization functions arising from the above parametrizations. A graph can be defined on the set of simplicial regions of linearity with respect to adjacency, and we further show that this graph is isomorphic to the graph with vertices given by the reduced expressions of the longest word of the Weyl group modulo commutation and edges given by long braid relations
Transitive Actions of Compact Groups and Topological Dimension

AbstractThere are many dimension functions defined on arbitrary topological spaces taking either a finite value or the value infinity. This paper defines a cardinal valued dimension function, dim. The Lie algebra L(G) of a compact group G is a weakly complete topological vector space. Quotient spaces of weakly complete spaces are weakly complete; the dimension of a weakly complete vector space is the linear dimension of its dual. Assume that a compact group G acts transitively on a given space X and that H is the isotropy group of the action at an arbitrary point; let L(G) and L(H) denote the Lie algebras of G, respectively, H. It is shown that dimX=dimL(G)/L(H). Moreover, such an X contains a space homeomorphic to [0,1]dimX; conversely, if X contains a homeomorphic copy of a cube [0,1]ℵ, then ℵ≤dimX. En route one establishes a good deal of information on the quotient spaces G/H; such information is of independent interest. Finally, these results are generalized to quotient spaces of locally compact groups. A generalization of a theorem of Iwasawa is instrumental; it is of independent interest as well
Transitive Subgroups of Primitive Permutation Groups

AbstractWe investigate the finite primitive permutation groups G which have a transitive subgroup containing no nontrivial subnormal subgroup of G. The conclusion is that such primitive groups are rather rare, and that their existence is intimately connected with factorisations of almost simple groups. A corollary is obtained on primitive groups which contain a regular subgroup. Heavily involved in our proofs are some new results on subgroups of simple groups which have orders divisible by various primes. For example, another corollary implies that for every simple group T apart from L3(3), U3(3), and L2(p) with p a Mersenne prime, there is a collection Π consisting of two or three odd prime divisors of |T|, such that if M is a subgroup of T of order divisible by every prime in Π, then |M| is divisible by all the prime divisors of |T|, and we obtain a classification of such subgroups M