Specht Modules Research Articles

A New Presentation for Specht Modules with Distinct Parts

We obtain a new presentation for Specht modules whose conjugate shapes have strictly decreasing parts by introducing a linear operator on the space generated by column tabloids. The generators of the presentation are column tabloids and the relations form a proper subset of the Garnir relations of Fulton. The results in this paper generalize earlier results of the authors and Stanley on Specht modules of staircase shape.

On the endomorphism algebra of Specht modules in even characteristic

Over fields of characteristic 2, Specht modules may decompose and there is no upper bound for the dimension of their endomorphism algebra. A classification of the (in)decomposable Specht modules and a closed formula for the dimension of their endomorphism algebra remain two important open problems in the area. In this paper, we introduce a novel description of the endomorphism algebra of the Specht modules and provide infinite families of Specht modules with one-dimensional endomorphism algebra.

Equivalence of v-decomposition matrices for blocks of Ariki-Koike algebras

We consider the representation theory of the Ariki-Koike algebra, a q-deformation of the group algebra of the complex reflection group Cr≀Sn. We examine blocks of the Ariki-Koike algebra. In particular, we prove a sufficient condition such that restriction of modules leads to a natural correspondence between the multipartitions of n whose Specht modules belong to a block B and those of n−δi(B) whose Specht modules belong to the block B′, obtained from B applying a Scopes' equivalence. This bijection gives us an equivalence for the v-decomposition numbers of the Ariki-Koike algebras.

Core blocks for Hecke algebras of type B and sign sequences

We consider the core blocks corresponding to the Hecke algebras of type B over a field of arbitrary characteristic. To each core block, we associate two non-negative integers which determine the indexing of the Specht modules and simple modules in the block, the weight of the block, the multicharge of the algebra (up to a shift) and the block decomposition matrix.

Specht modules for partitions (k,3,3) have no one-dimensional summand

Specht modules for partitions (k,3,3) have no one-dimensional summand

Categorical braid group actions and cactus groups

Let g be a semisimple simply-laced Lie algebra of finite type. Let C be an abelian categorical representation of the quantum group Uq(g) categorifying an integrable representation V. The Artin braid group B of g acts on Db(C) by Rickard complexes, providing a triangulated equivalenceΘw0:Db(Cμ)→Db(Cw0(μ)) where μ is a weight of V, and Θw0 is a positive lift of the longest element of the Weyl group.We prove that this equivalence is t-exact up to shift when V is isotypic, generalising a fundamental result of Chuang and Rouquier in the case g=sl2. For general V, we prove that Θw0 is a perverse equivalence with respect to a Jordan-Hölder filtration of C.Using these results we construct, from the action of B on V, an action of the cactus group on the crystal of V. This recovers the cactus group action on V defined via generalised Schützenberger involutions, and provides a new connection between categorical representation theory and crystal bases. We also use these results to give new proofs of theorems of Berenstein-Zelevinsky, Rhoades, and Stembridge regarding the action of symmetric group on the Kazhdan-Lusztig basis of its Specht modules.

Positivity and Web Bases for Specht Modules of Hecke Algebras

Abstract We show that the transition matrix from the standard basis to the web basis for a Specht module of the Hecke algebra is unitriangular and satisfies a strong positivity property whenever the Specht module is labeled by a partition with at most two parts. This generalizes results of Russell–Tymoczko and Rhoades.

Stable Centres of Iwahori-Hecke Algebras of Type A

A celebrated result of Farahat and Higman constructs an algebra FH which “interpolates” the centres \(Z(\mathbb {Z}S_{n})\) of group algebras of the symmetric groups Sn. We extend these results from symmetric group algebras to type A Iwahori-Hecke algebras, Hn(q). In particular, we explain how to construct an algebra FHq “interpolating” the centres Z(Hn(q)). We prove that FHq is isomorphic to \(\mathcal {R}[q,q^{-1}] \otimes _{\mathbb {Z}} {\Lambda }\) (where \(\mathcal {R}\) is the ring of integer-valued polynomials, and Λ is the ring of symmetric functions). The isomorphism can be described as “evaluation at Jucys-Murphy elements”, leading to a proof of a conjecture of Francis and Wang. This yields character formulae for the Geck-Rouquier basis of Z(Hn(q)) when acting on Specht modules.

Specht module branching rules for wreath products of symmetric groups

We review a class of modules for the wreath product S(m) wr S(n) of two symmetric groups which are analogous to the Specht modules of the symmetric group, and prove a pair of branching rules for this family of modules. These branching rules describe the behaviour of these wreath product Specht modules under restriction to the wreath products S(m-1) wr S(n) and S(m) wr S(n-1). In particular, we see that these restrictions of wreath product Specht modules have Specht module filtrations, and we obtain combinatorial interpretations of the multiplicities in these filtrations.

Graded Specht Modules as Bernstein–Zelevinsky Derivatives of the RSK Model

AbstractWe clarify the links between the graded Specht construction of modules over cyclotomic Hecke algebras and the Robinson-Schensted-Knuth (RSK) construction for quiver Hecke algebras of type $A$, which was recently imported from the setting of representations of $p$-adic groups. For that goal we develop a theory of crystal derivative operators on quiver Hecke algebra modules that categorifies the Berenstein–Zelevinsky strings framework on quantum groups and generalizes a graded variant of the classical Bernstein–Zelevinsky derivatives from the $p$-adic setting. Graded cyclotomic decomposition numbers are shown to be a special subfamily of the wider concept of RSK decomposition numbers.

A web basis of invariant polynomials from noncrossing partitions

The irreducible representations of symmetric groups can be realized as certain graded pieces of invariant rings, equivalently as global sections of line bundles on partial flag varieties. There are various ways to choose useful bases of such Specht modules Sλ. Particularly powerful are web bases, which make important connections with cluster algebras and quantum link invariants. Unfortunately, web bases are only known in very special cases—essentially, only the cases λ=(d,d) and λ=(d,d,d). Building on work of B. Rhoades (2017), we construct an apparent web basis of invariant polynomials for the 2-parameter family of Specht modules with λ of the form (d,d,1ℓ). The planar diagrams that appear are noncrossing set partitions, and we thereby obtain geometric interpretations of earlier enumerative results in combinatorial dynamics.

A simplified presentation of Specht modules

Fulton and Kraskiewicz gave a presentation of Specht modules as a quotient of the space of column tabloids by dual Garnir relations. We simplify this presentation by showing that it can be generated by a single relation for each pair of columns of a tableau with ordered columns, thereby significantly reducing the number of generators given in the original construction. Our presentation applies to all Specht modules, and is of a similar nature to a recent result by Friedmann-Hanlon-Stanley-Wachs that applies to staircase partitions. We show that our presentation implies the Friedmann-Hanlon-Stanley-Wachs presentation.

Elementary construction of minimal free resolutions of the Specht ideals of shapes (n − 2,2) and (d,d,1)

For a partition [Formula: see text] of [Formula: see text], let [Formula: see text] be the ideal of [Formula: see text] generated by all Specht polynomials of shape [Formula: see text]. We assume that [Formula: see text]. Then [Formula: see text] is Gorenstein, and [Formula: see text] is a Cohen–Macaulay ring with a linear free resolution. In this paper, we construct minimal free resolutions of these rings. Zamaere et al. [Jack polynomials as fractional quantum Hall states and the Betti numbers of the [Formula: see text]-equals ideal, Commun. Math. Phys. 330 (2014) 415–434] already studied minimal free resolutions of [Formula: see text], which are also Cohen–Macaulay, using highly advanced technique of the representation theory. However, we only use the basic theory of Specht modules, and explicitly describe the differential maps.

Schur functions in noncommuting variables

In 2004 Rosas and Sagan asked whether there was a way to define a basis in the algebra of symmetric functions in noncommuting variables, NCSym, having properties analogous to the classical Schur functions. This was because they had constructed a partial such set that was not a basis. We answer their question by defining Schur functions in noncommuting variables using a noncommutative analogue of the Jacobi-Trudi determinant. Our Schur functions in NCSym map to classical Schur functions under commutation, and a subset of them indexed by set partitions forms a basis for NCSym. Amongst other properties, Schur functions in NCSym also satisfy a noncommutative analogue of the product rule for classical Schur functions in terms of skew Schur functions.We also show how Schur functions in NCSym are related to Specht modules, and naturally refine the Rosas-Sagan Schur functions. Moreover, by generalizing Rosas-Sagan Schur functions to skew Schur functions in the natural way, we prove noncommutative analogues of the Littlewood-Richardson rule and coproduct rule for them. Finally, we relate our functions to noncommutative symmetric functions by proving a subset of our functions are natural extensions of noncommutative ribbon Schur functions, and immaculate functions indexed by integer partitions.

On the denominators of Young's seminormal basis

We study the seminormal basis {ft} for the Specht modules of the Iwahori-Hecke algebra Hn(q) of type An−1. We focus on the base change coefficients between the seminormal basis {ft} and Murphys' standard basis {xt} with emphasis on the denominators of these coefficients. In certain important cases we obtain simple formulas for these coefficients involving hook lengths. Even for general standard tableaux we obtain new formulas. On the way we prove a new result about submodules of the restricted Specht module at root of unity.

MODUL WEYL SEBAGAI REPRESENTASI GRUP SIMETRI DARI PEWARNAAN SUATU GRAF

This research identifies the properties of Weyl module which had some similar properties with Specht module. Specht module as an irreducible representation of permutation module for some partition

Positive Jantzen sum formulas for cyclotomic Hecke algebras

This paper proves a “positive” Jantzen sum formula for the Specht modules of the cyclotomic Hecke algebras of type A and uses it to obtain new bounds on decomposition numbers. Quite remarkably, our results are proved entirely inside the cyclotomic Hecke algebras. Our positive sum formula shows that, in the Grothendieck group, the Jantzen sum formula can be written as an explicit non-negative linear combination of modules [E^{varvec{nu }}_{f,e}], which are the modular reductions of simple modules of related Hecke algebras in characteristic zero. The coefficient of [E^{varvec{nu }}_{f,e}] in the sum formula is determined by the graded decomposition numbers in characteristic zero, which are known, and by the characteristic of the field. As a consequence we give an explicit upper bound for the decomposition numbers in characteristic p>0 in terms of linear combinations of decomposition numbers of a cyclotomic Hecke algebra at ep^rth roots of unity in characteristic zero, for rge 0. Finally, we prove a new and more elegant “classical” Jantzen sum formula for these algebras.

An equivariant Hochster's formula for Sn$\mathfrak {S}_n$‐invariant monomial ideals

Let $R=\Bbbk[x_1,\dots,x_n]$ be a polynomial ring over a field $\Bbbk$ and let $I\subset R$ be a monomial ideal preserved by the natural action of the symmetric group $\mathfrak S_n$ on $R$. We give a combinatorial method to determine the $\mathfrak S_n$-module structure of $\mathrm{Tor}_i(I,\Bbbk)$. Our formula shows that $\mathrm{Tor}_i(I,\Bbbk)$ is built from induced representations of tensor products of Specht modules associated to hook partitions, and their multiplicities are determined by topological Betti numbers of certain simplicial complexes. This result can be viewed as an $\mathfrak S_n$-equivariant analogue of Hochster's formula for Betti numbers of monomial ideals. We apply our results to determine extremal Betti numbers of $\mathfrak S_n$-invariant monomial ideals, and in particular recover formulas for their Castelnuovo--Mumford regularity and projective dimension. We also give a concrete recipe for how the Betti numbers change as we increase the number of variables, and in characteristic zero (or $>n$) we compute the $\mathfrak S_n$-invariant part of $\mathrm{Tor}_i(I,\Bbbk)$ in terms of $\mathrm{Tor}$ groups of the unsymmetrization of $I$.

Kronecker positivity and 2-modular representation theory

This paper consists of two prongs. Firstly, we prove that any Specht module labelled by a 2-separated partition is semisimple and we completely determine its decomposition as a direct sum of graded simple modules. Secondly, we apply these results and other modular representation theoretic techniques on the study of Kronecker coefficients and hence verify Saxl’s conjecture for several large new families of partitions. In particular, we verify Saxl’s conjecture for all irreducible characters of S n \mathfrak {S}_n which are of 2-height zero.

Decomposable Specht Modules Indexed by Bihooks II

Previously, the last two authors found large families of decomposable Specht modules labelled by bihooks, over the Iwahori–Hecke algebra of type B. In most cases we conjectured that these were the only decomposable Specht modules labelled by bihooks, proving it in some instances. Inspired by a recent semisimplicity result of Bowman, Bessenrodt and the third author, we look back at our decomposable Specht modules and show that they are often either semisimple, or very close to being so. We obtain their exact structure and composition factors in these cases. In the process, we determine the graded decomposition numbers for almost all of the decomposable Specht modules indexed by bihooks.