Showing papers on "Symmetric group published in 2014"
Book•
12 Mar 2014
TL;DR: In this paper, the standard basis of the specht module is defined, and the branching theorem and branching theorem can be used to obtain irreducible representations of the symmetric group.
Abstract: Background from representation theory.- The symmetric group.- Diagrams, tableaux and tabloids.- Specht modules.- Examples.- The character table of .- The garnir relations.- The standard basis of the specht module.- The branching theorem.- p-regular partitions.- The irreducible representations of .- Composition factors.- Semistandard homomorphisms.- Young's rule.- Sequences.- The Littlewood-richardson rule.- A specht series for M?.- Hooks and skew-hooks.- The determinantal form.- The hook formula for dimensions.- The murnaghan-nakayama rule.- Binomial coefficients.- Some irreducible specht modules.- On the decomposition matrices of .- Young's orthogonal form.- Representations of the general linear group.
884 citations
TL;DR: The Noetherian property of FI-modules was shown in this paper, where it was shown that for any sub-FI-module of a finitely generated FI-module, the representation stability of the corresponding sequence of Sn-representations is guaranteed.
Abstract: FI-modules were introduced by the first three authors to encode sequences of representations of symmetric groups. Over a field of characteristic 0, finite generation of an FI-module implies representation stability for the corresponding sequence of Sn ‐representations. In this paper we prove the Noetherian property for FI-modules over arbitrary Noetherian rings: any sub-FI-module of a finitely generated FI-module is finitely generated. This lets us extend many results to representations in positive characteristic, and even to integral coefficients. We focus on three major applications of the main theorem: on the integral and mod p cohomology of configuration spaces; on diagonal coinvariant algebras in positive characteristic; and on an integral version of Putman’s central stability for homology of congruence subgroups. 20B30; 20C32
132 citations
TL;DR: In this paper, the role of the symmetric group is played by the general linear groups and the symplectic groups over finite rings and prove basic structural properties such as Noetherianity.
Abstract: We construct analogues of FI-modules where the role of the symmetric group is played by the general linear groups and the symplectic groups over finite rings and prove basic structural properties such as Noetherianity. Applications include a proof of the Lannes--Schwartz Artinian conjecture in the generic representation theory of finite fields, very general homological stability theorems with twisted coefficients for the general linear and symplectic groups over finite rings, and representation-theoretic versions of homological stability for congruence subgroups of the general linear group, the automorphism group of a free group, the symplectic group, and the mapping class group.
99 citations
TL;DR: In this article, it was shown that 2n-and (2mn+1)-core partitions correspond naturally to dominant alcove in the m-Shi arrangement of type C"n, generalizing a result of Fishel-Vazirani for type A.
Abstract: An n-core partition is an integer partition whose Young diagram contains no hook lengths equal to n. We consider partitions that are simultaneously a-core and b-core for two relatively prime integers a and b. These are related to abacus diagrams and the combinatorics of the affine symmetric group (type A). We observe that self-conjugate simultaneous core partitions correspond to the combinatorics of type C, and use abacus diagrams to unite the discussion of these two sets of objects. In particular, we prove that 2n- and (2mn+1)-core partitions correspond naturally to dominant alcoves in the m-Shi arrangement of type C"n, generalizing a result of Fishel-Vazirani for type A. We also introduce a major index statistic on simultaneous n- and (n+1)-core partitions and on self-conjugate simultaneous 2n- and (2n+1)-core partitions that yield q-analogs of the Coxeter-Catalan numbers of type A and type C. We present related conjectures and open questions on the average size of a simultaneous core partition, q-analogs of generalized Catalan numbers, and generalizations to other Coxeter groups. We also discuss connections with the cyclic sieving phenomenon and q,t-Catalan numbers.
89 citations
TL;DR: In this paper, the authors developed a framework for studying sequences of representations of any of the three families of classical Weyl groups, extending work of Church, Ellenberg, Farb, and Nagpal [7,8] on the symmetric groups S n to the signed permutation groups B n and D n.
72 citations
TL;DR: The primitive finite permutation groups containing a cycle were classified in this paper, and only the alternating and symmetric groups contained a cycle fixing at least three points, which removed a primality condition from a classical theorem of Jordan.
Abstract: The primitive finite permutation groups containing a cycle are classified. Of these, only the alternating and symmetric groups contain a cycle fixing at least three points. This removes a primality condition from a classical theorem of Jordan. Some applications to monodromy groups are given, and the contributions of Jordan and Marggraff to this topic are briefly discussed.
10.1017/S000497271300049X
69 citations
Posted Content•
TL;DR: In this paper, the authors generalize this setup to rational parking functions indexed by a pair (a, b) of coprime positive integers and show that these parking functions, which are counted by b^(a-1), carry a permutation representation of S_a in which the number of orbits is a rational Catalan number.
Abstract: The classical parking functions, counted by the Cayley number (n+1)^(n-1), carry a natural permutation representation of the symmetric group S_n in which the number of orbits is the n'th Catalan number. In this paper, we will generalize this setup to rational parking functions indexed by a pair (a,b) of coprime positive integers. We show that these parking functions, which are counted by b^(a-1), carry a permutation representation of S_a in which the number of orbits is a rational Catalan number. We compute the Frobenius characteristic of the S_a-module of (a,b)-parking functions. Next we propose a combinatorial formula for a q-analogue of the rational Catalan numbers and relate this formula to a new combinatorial model for q-binomial coefficients. Finally, we discuss q,t-analogues of rational Catalan numbers and parking functions (generalizing the shuffle conjecture for the classical case) and present several conjectures.
68 citations
TL;DR: In this article, the authors proposed a method of interpolating representation categories of various algebras containing symmetric groups to complex values of n. This construction was generalized by F.P. Knop to the case of wreath products with a finite group.
Abstract: P. Deligne defined interpolations of the tensor category of representations of the symmetric group S
n
to complex values of n. Namely, he defined tensor categories Rep(S
t
) for any complex t. This construction was generalized by F. Knop to the case of wreath products of S
n
with a finite group. Generalizing these results, we propose a method of interpolating representation categories of various algebras containing S
n
(such as degenerate affine Hecke algebras, symplectic reflection algebras, rational Cherednik algebras, etc.) to complex values of n. We also define the group algebra of S
n
for complex n, study its properties, and propose a Schur-Weyl duality for Rep(S
t
).
54 citations
TL;DR: In this article, the authors studied the inductive blockwise Alperin weight conjecture for groups of Lie type in the generic case of abelian Sylow l-subgroups.
54 citations
TL;DR: In this article, a quasipolynomial upper bound on the diameter of the Cayley graph of a finite group G and a set of generators was given, where G =Sym(n) or G=Alt(n), where the implied constants are absolute.
Abstract: Given a finite group G and a set A of generators, the diameter
diam(G(G,A)) of the Cayley graph G(G,A) is the smallest l such that every element of G can be expressed as a word of length at most l in A?A -1 . We are concerned with bounding diam(G):=max A diam(G(G,A)) .
It has long been conjectured that the diameter of the symmetric group of degree n is polynomially bounded in n , but the best previously known upper bound was exponential in nlogn - - - - - v . We give a quasipolynomial upper bound, namely,
diam(G)=exp(O((logn) 4 loglogn))=exp((loglog|G|) O(1) ) for G=Sym(n) or G=Alt(n) , where the implied constants are absolute. This addresses a key open case of Babai�s conjecture on diameters of simple groups. By a result of Babai and Seress (1992), our bound also implies a quasipolynomial upper bound on the diameter of all transitive permutation groups of degree n .
TL;DR: The notion of L-subgyrogroup was introduced in this article to partition the symmetric group of a gyrogroup into left cosets, where the order of a subgroup of the group is the same as that of the whole group.
Abstract: We extend well-known results in group theory to gyrogroups, especially the isomorphism theorems. We prove that an arbitrary gyrogroup $G$ induces the gyrogroup structure on the symmetric group of $G$ so that Cayley's Theorem is obtained. Introducing the notion of L-subgyrogroups, we show that an L-subgyrogroup partitions $G$ into left cosets. Consequently, if $H$ is an L-subgyrogroup of a finite gyrogroup $G$, then the order of $H$ divides the order of $G$.
01 Jan 2014
TL;DR: The notion of L-subgyrogroup was introduced in this article to partition the symmetric group of a gyrogroup into left cosets, where the order of a subgroup of the group is the same as that of the whole group.
Abstract: We extend well-known results in group theory to gyrogroups, especially the isomorphism theorems. We prove that an arbitrary gyrogroup $G$ induces the gyrogroup structure on the symmetric group of $G$ so that Cayley's Theorem is obtained. Introducing the notion of L-subgyrogroups, we show that an L-subgyrogroup partitions $G$ into left cosets. Consequently, if $H$ is an L-subgyrogroup of a finite gyrogroup $G$, then the order of $H$ divides the order of $G$.
TL;DR: In this article, it was shown that the Chow ring of the classifying space BG depends on the base field, even for fields containing the algebraic closure of Q. This makes it possible to ask when the motive of BG is mixed Tate, which is equivalent to the motivic Kunneth property.
Abstract: We give the first examples of finite groups G such that the Chow ring of the classifying space BG depends on the base field, even for fields containing the algebraic closure of Q. As a tool, we give several characterizations of the varieties which satisfy Kunneth properties for Chow groups or motivic homology.
We define the (compactly supported) motive of a quotient stack in Voevodsky's derived category of motives. This makes it possible to ask when the motive of BG is mixed Tate, which is equivalent to the motivic Kunneth property. We prove that BG is mixed Tate for various "well-behaved" finite groups G, such as the finite general linear groups in cross-characteristic and the symmetric groups.
TL;DR: In this paper, it was shown that every finite group G can be realized as the group of self-homotopy equivalences of infinitely many elliptic spaces X. To construct those spaces, a new technique was introduced, which leads, for example, to the existence of infinite many inflexible manifolds.
Abstract: We prove that every finite group G can be realized as the group of self-homotopy equivalences of infinitely many elliptic spaces X. To construct those spaces we introduce a new technique which leads, for example, to the existence of infinitely many inflexible manifolds. Further applications to representation theory will appear in a separate paper.
TL;DR: In this paper, a new approach was proposed to study the Kronecker coefficients by using the Schur-Weyl duality between the symmetric group and the partition algebra, which leads to a uniform description of the reduced Kroneck coefficients when one of the indexing partitions is a hook or a two-part partition.
Abstract: We propose a new approach to study the Kronecker coefficients by using the Schur-Weyl duality between the symmetric group and the partition algebra. We explain the limiting behaviour and associated bounds in the context of the partition algebra. Our analysis leads to a uniform description of the reduced Kronecker coefficients when one of the indexing partitions is a hook or a two-part partition.
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01 Jan 2014
TL;DR: In this paper, a generalization of this interpretation is given for the local h-polynomial of the rth edgewise subdivision of the barycentric subdivision of a simplex with n vertices.
Abstract: The coefficients of the localh-polynomial of the barycentric subdivision of the simplex with n vertices are known to count derangements in the symmetric group Sn by the number of excedances. A generalization of this interpretation is given for the local h-polynomial of the rth edgewise subdivision of the barycentric subdivision of the simplex. This polynomial is shown to be -nonnegative and a combinatorial interpretation to the corresponding -coefficients is provided. The new combinatorial interpretations involve the notions of flag excedance and descent in the wreath product Zr ≀Sn. A related result on the derangement polynomial for Zr ≀Sn, studied by Chow and Mansour, is also derived from results of Linusson, Shareshian and Wachs on the homology of Rees products of posets.
TL;DR: In this article, it was shown that a weighted Bergman space on the polydisc splits up into the orthogonal direct sum of subspaces, each corresponding to an irreducible representation of the symmetric group.
TL;DR: In this paper, the authors consider generalizations of Schutzenberger's promotion operator on the set of linear extensions of a finite poset of size n. This gives rise to a strongly connected graph on $\mathcal{L}$.
Abstract: We consider generalizations of Schutzenberger's promotion operator on the set $\mathcal{L}$ of linear extensions of a finite poset of size n. This gives rise to a strongly connected graph on $\mathcal{L}$ . By assigning weights to the edges of the graph in two different ways, we study two Markov chains, both of which are irreducible. The stationary state of one gives rise to the uniform distribution, whereas the weights of the stationary state of the other have a nice product formula. This generalizes results by Hendricks on the Tsetlin library, which corresponds to the case when the poset is the anti-chain and hence $\mathcal{L}=S_{n}$ is the full symmetric group. We also provide explicit eigenvalues of the transition matrix in general when the poset is a rooted forest. This is shown by proving that the associated monoid is $\mathcal {R}$ -trivial and then using Steinberg's extension of Brown's theory for Markov chains on left regular bands to $\mathcal {R}$ -trivial monoids.
TL;DR: In this article, all the elliptic fibrations with section on the Kummer surface X of the Jacobian of a very general curve C of genus 2 over an algebraically closed field of characteristic 0, modulo the automorphism group of X and the symmetric group on the Weierstrass points of C, were described.
Abstract: We describe all the elliptic fibrations with section on the Kummer surface X of the Jacobian of a very general curve C of genus 2 over an algebraically closed field of characteristic 0, modulo the automorphism group of X and the symmetric group on the Weierstrass points of C. In particular, we compute elliptic parameters and Weierstrass equations for the 25 different fibrations and analyze the reducible fibers and Mordell-Weil lattices. This answers completely a question posed by Kuwata and Shioda in 2008.
TL;DR: A simple product formula is obtained for the exponential generating function of factorizations of Coxeter elements in well-generated complex reflection groups into products of reflections, which is expressed uniformly in terms of natural parameters of the group.
Abstract: In this paper, we count factorizations of Coxeter elements in well-generated complex reflection groups into products of reflections. We obtain a simple product formula for the exponential generating function of such factorizations, which is expressed uniformly in terms of natural parameters of the group. In the case of factorizations of minimal length, we recover a formula due to P. Deligne, J. Tits and D. Zagier in the real case and to D. Bessis in the complex case. For the symmetric group, our formula specializes to a formula of D. M. Jackson.
TL;DR: In this article, a connection of HOMFLY polynomials with Hurwitz covers and represent a generating function for the HOG polynomial of a given knot in all representations as Hurwitz partition function is discussed.
Abstract: We discuss a connection of HOMFLY polynomials with Hurwitz covers and represent a generating function for the HOMFLY polynomial of a given knot in all representations as Hurwitz partition function, i.e. the dependence of the HOMFLY polynomials on representation R is naturally captured by symmetric group characters (cut-and-join eigenvalues). The genus expansion and the loop expansion through Vassiliev invariants explicitly demonstrate this phenomenon. We study the genus expansion and discuss its properties. We also consider the loop expansion in details. In particular, we give an algorithm to calculate Vassiliev invariants, give some examples and discuss relations among Vassiliev invariants. Then we consider superpolynomials for torus knots defined via double affine Hecke algebra. We claim that the superpolynomials are not functions of Hurwitz type: symmetric group characters do not provide an adequate linear basis for their expansions. Deformation to superpolynomials is, however, straightforward in the multiplicative basis: the Casimir operators are beta-deformed to Hamiltonians of the Calogero–Moser–Sutherland system. Applying this trick to the genus and Vassiliev expansions, we observe that the deformation is fully straightforward only for the thin knots. Beyond the family of thin knots additional algebraically independent terms appear in the Vassiliev expansions. This can suggest that the superpolynomials do in fact contain more information about knots than the colored HOMFLY and Kauffman polynomials.
TL;DR: The Little map and the Edelman-Greene insertion algorithm were used for enumerating the reduced decompositions of an element of the symmetric group in this paper, and the results about the Little map factors were established.
Abstract: The Little map and the Edelman---Greene insertion algorithm, a generalization of the Robinson---Schensted correspondence, are both used for enumerating the reduced decompositions of an element of the symmetric group. We show the Little map factors through Edelman---Greene insertion and establish new results about each map as a consequence. In particular, we resolve some conjectures of Lam and Little.
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TL;DR: In this paper, a Ricci curvature lower bound for some classical examples of random walks, namely, a chain on a slice of the n-dimensional discrete cube (the so-called Bernoulli-Laplace model) and the random transposition shuffle of the symmetric group of permutations on n letters, was derived.
Abstract: We calculate a Ricci curvature lower bound for some classical examples of random walks, namely, a chain on a slice of the n-dimensional discrete cube (the so-called Bernoulli-Laplace model) and the random transposition shuffle of the symmetric group of permutations on n letters.
08 Jul 2014
TL;DR: The notion of uniform minimality to direct products of automata is generalized and the non-trivial connection between complexity of boolean operations and group theory is established.
Abstract: The quotient complexity of a regular language L is the number of left quotients of L, which is the same as the state complexity of L. Suppose that L and L′ are binary regular languages with quotient complexities m and n, and that the subgroups of permutations in the transition semigroups of the minimal deterministic automata accepting L and L′ are the symmetric groups S m and S n of degrees m and n, respectively. Denote by ∘ any binary boolean operation that is not a constant and not a function of one argument only. For m,n ≥ 2 with \((m,n)
ot \in \{(2,2),(3,4),(4,3),(4,4)\}\) we prove that the quotient complexity of L ∘ L′ is mn if and only either (a) \(m
ot= n\) or (b) m = n and the bases (ordered pairs of generators) of S m and S n are not conjugate. For (m,n) ∈ {(2,2),(3,4),(4,3),(4,4)} we give examples to show that this need not hold. In proving these results we generalize the notion of uniform minimality to direct products of automata. We also establish a non-trivial connection between complexity of boolean operations and group theory.
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TL;DR: The W -set of an element of a weak order poset is useful in the cohomological study of the closures of spherical subgroups in generalized flag varieties and the maximal chains of an arbitrary lower order ideal in any of these three posets are characterized.
Abstract: The W-set of an element of a weak order poset is useful in the cohomological study of the closures of spherical subgroups in generalized flag varieties. We explicitly describe in a purely combinatorial manner the W-sets of the weak order posets of three different sets of involutions in the symmetric group, namely, the set of all involutions, the set of all fixed point free involutions, and the set of all involutions with signed fixed points (or "clans"). These distinguished sets of involutions parameterize Borel orbits in the classical symmetric spaces associated to the general linear group. In particular, we give a complete characterization of the maximal chains of an arbitrary lower order ideal in any of these three posets.
TL;DR: In this paper, a general description of the moment cone associated with an arbitrary finite-dimensional unitary representation of a compact, connected Lie group in terms of finitely many linear inequalities is given.
Abstract: We give a general description of the moment cone associated with an arbitrary finite-dimensional unitary representation of a compact, connected Lie group in terms of finitely many linear inequalities. Our method is based on combining differential-geometric arguments with a variant of Ressayre's notion of a dominant pair. As applications, we obtain generalizations of Horn's inequalities to arbitrary representations, new inequalities for the one-body quantum marginal problem in physics, which concerns the asymptotic support of the Kronecker coefficients of the symmetric group, and a geometric interpretation of the Howe-Lee-Tan-Willenbring invariants for the tensor product algebra.
TL;DR: The representation theory of locally compact infinite-dimensional group GLB of almost upper-triangular infinite matrices over the finite field with q elements was studied in this article. But it was not shown that the representation theory can be applied to finite groups.
TL;DR: Zhang et al. as discussed by the authors showed that a factorization formula for the cluster variables arising from a large class of mutation sequences (called τ-mutation sequences) is proven; this factorization also gives a recursion on the cluster variable produced by such sequences.
Abstract: Bipartite, periodic, planar graphs known as brane tilings can be associated to a large class of quivers. This paper will explore new algebraic properties of the well-studied del Pezzo 3 (dP3) quiver and geometric properties of its corresponding brane tiling. In particular, a factorization formula for the cluster variables arising from a large class of mutation sequences (called τ-mutation sequences) is proven; this factorization also gives a recursion on the cluster variables produced by such sequences. We can realize these sequences as walks in a triangular lattice using a correspondence between the generators of the affine symmetric group and the mutations which generate τ-mutation sequences. Using this bijection, we obtain explicit formulae for the cluster that corresponds to a specific alcove in the lattice. With this lattice visualization in mind, we then express each cluster variable produced in a τ-mutation sequence as the sum of weighted perfect matchings of a new family of subgraphs of the dP3 brane tiling, which we call Aztec castles. Our main result generalizes previous work on a certain mutation sequence on the dP3 quiver in Zhang (2012 Cluster Variables and Perfect Matchings of Subgraphs of the dP3 Lattice http://www.math.umn.edu/~/REU/Zhang2012.pdf), and forms part of the emerging story in combinatorics and theoretical high energy physics relating cluster variables to subgraphs of the associated brane tiling.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to 'Cluster algebras in mathematical physics'.
TL;DR: In this paper, a new product for permutation groups is introduced, which takes as input two permutation group, M and N, and produces an infinite group M [X] N which carries many of the permutational properties of M under mild conditions on M and n. The product provides a general way to build exotic examples of non-discrete, simple, totally disconnected, locally compact, compactly generated topological groups from discrete groups.
Abstract: We introduce a new product for permutation groups It takes as input two permutation groups, M and N, and produces an infinite group M [X] N which carries many of the permutational properties of M Under mild conditions on M and N the group M [X] N is simple
As a permutational product, its most significant property is the following: M [X] N is primitive if and only if M is primitive but not regular, and N is transitive Despite this remarkable similarity with the wreath product in product action, M [X] N and M Wr N are thoroughly dissimilar The product provides a general way to build exotic examples of non-discrete, simple, totally disconnected, locally compact, compactly generated topological groups from discrete groups
We use this to solve a well-known open problem from topological group theory, by obtaining the first construction of uncountably many pairwise non-isomorphic simple topological groups that are totally disconnected, locally compact, compactly generated and non-discrete The groups we construct all contain the same compact open subgroup
To build the product, we describe a group U(M,N) that acts on an edge-transitive biregular tree T This group has a natural universal property and is analogous to the iconic universal group construction of M Burger and S Mozes for locally finite regular trees