Specht Modules Research Articles

Character Polynomials and the Restriction Problem

Character polynomials are used to study the restriction of a polynomial representation of a general linear group to its subgroup of permutation matrices. A simple formula is obtained for computing inner products of class functions given by character polynomials. Character polynomials for symmetric and alternating tensors are computed using generating functions with Eulerian factorizations. These are used to compute character polynomials for Weyl modules, which exhibit a duality. By taking inner products of character polynomials for Weyl modules and character polynomials for Specht modules, stable restriction coefficients are easily computed. Generating functions of dimensions of symmetric group invariants in Weyl modules are obtained. Partitions with two rows, two columns, and hook partitions whose Weyl modules have non-zero vectors invariant under the symmetric group are characterized. A reformulation of the restriction problem in terms of a restriction functor from the category of strict polynomial functors to the category of finitely generated FI-modules is obtained.

Vandermonde Varieties, Mirrored Spaces, and the Cohomology of Symmetric Semi-algebraic Sets

Let $\mathrm{R}$ be a real closed field. We prove that for each fixed $\ell, d \geq 0$, there exists an algorithm that takes as input a quantifier-free first order formula $\Phi$ with atoms $P=0, P > 0, P < 0 \text{ with } P \in \mathcal{P} \subset \mathrm{D}[X_1,\ldots,X_k]^{\mathfrak{S}_k}_{\leq d}$, where $\mathrm{D}$ is an ordered domain contained in $\mathrm{R}$, and computes the ranks of the first $(\ell+1)$ cohomology groups, of the symmetric semi-algebraic set defined by $\Phi$. The complexity of this algorithm (measured by the number of arithmetic operations in $\mathrm{D}$) is bounded by a \emph{polynomial} in $k$ and $\mathrm{card}(\mathcal{P})$ (for fixed $d$ and $\ell$). This result contrasts with the $\mathbf{PSPACE}$-hardness of the problem of computing just the zero-th Betti number (i.e. the number of semi-algebraically connected components) in the general case for $d \geq 2$ (taking the ordered domain $\mathrm{D}$ to be equal to $\mathbb{Z}$). The above algorithmic result is built on new representation theoretic results on the cohomology of symmetric semi-algebraic sets. We prove that the Specht modules corresponding to partitions having long lengths cannot occur with positive multiplicity in the isotypic decompositions of low dimensional cohomology modules of closed semi-algebraic sets defined by symmetric polynomials having small degrees. This result generalizes prior results obtained by the authors giving restrictions on such partitions in terms of their ranks, and is the key technical tool in the design of the algorithm mentioned in the previous paragraph.

The image of the Specht module under the inverse Schur functor in arbitrary characteristic

This paper gives a necessary and sufficient condition for the image of the Specht module under the inverse Schur functor to be isomorphic to the dual Weyl module in characteristic 2, and gives an elementary proof that this isomorphism holds in all cases in all other characteristics. These results are new in characteristics 2 and 3. We deduce some new examples of indecomposable Specht modules in characteristic 2. When the isomorphism does not hold, the dual Weyl module is still a quotient of the image of the Specht module, and we prove some additional results: we demonstrate that the image need not have a filtration by dual Weyl modules, we bound the dimension of the kernel of the quotient map, and we give some explicit descriptions for particular cases. Our method is to view the Specht and dual Weyl modules as quotients of suitable exterior powers by the Garnir relations.

Young's seminormal basis vectors and their denominators

We study Young's seminormal basis vectors of the dual Specht modules of the symmetric group, indexed by a certain class of standard tableaux, and their denominators. These vectors include those whose denominators control the splitting of the canonical morphism Δ(λ+μ)→Δ(λ)⊗Δ(μ) over Z(p), where Δ(ν) is the Weyl module of the classical Schur algebra labelled by ν.

Modular Irreducible Representations of the FpW4-Submodules ,()pFNof the Modules ,()pFMas Linear Codes, where W4is the Weyl Group of Type B4

The modular representations of the FpWn-Specht modules( , )KSas linear codes is given in our paper [6], and the modular irreducible representations of the FpW4-submodules( , )pFNof the Specht modules pFS ( , )as linear codes where W4is the Weyl group of type B4is given in our paper [5]. In this paper we are concerning of finding the linear codes of the representations of the irreducible FpW4-submodules( , )pFNof the FpW4-modules( , )pFMfor each pair of partitions( , )of a positive integer n4, where FpGF(p) is the Galois field (finite field) of order p, and pis a prime number greater than or equal to 3. We will find in this paper a generator matrix of a subspace((2,1),(1))()pU representing the irreducible FpW4-submodules((2,1),(1))pFNof the FpW4-modules((2,1),(1))pF Mand give the linear code of ((2,1),(1))()pU for each prime number p greater than or equal to 3. Then we will give the linear codes of all the subspaces( , )()pUfor all pair of partitions( , )of a positive integer n4, and for each prime number p greater than or equal to 3.We mention that some of the ideas of this work in this paper have been influenced by that of Adalbert Kerber and Axel Kohnert [13], even though that their paper is about the symmetric group and this paper is about the Weyl groups of type Bn

Periodicity in the cohomology of finite general linear groups via $q$-divided powers

We show that $\bigoplus_{n \ge 0} {\mathrm H}^t({\bf GL}_n({\bf F}_q), {\bf F}_\ell)$ canonically admits the structure of a module over the $q$-divided power algebra (assuming $q$ is invertible in ${\bf F}_{\ell}$), and that, as such, it is free and (for $q \neq 2$) generated in degrees $\le t$. As a corollary, we show that the cohomology of a finitely generated ${\bf VI}$-module in non-describing characteristic is eventually periodic in $n$. We apply this to obtain a new result on the cohomology of unipotent Specht modules.

On The Schaper Numbers Of Partitions

Abstract One of the most useful tools for calculating the decomposition numbers of the symmetric group is Schaper’s sum formula. The utility of this formula for a given Specht module can be improved by knowing the Schaper number of the corresponding partition. Fayers gives a characterization of those partitions whose Schaper number is at least two. In this paper, we shall demonstrate how this knowledge can be used to calculate some decomposition numbers before extending this result with the hope of allowing more decomposition numbers to be calculated in the future. For p = 2 we shall give a complete characterization of partitions whose Schaper number is at least three, and those whose Schaper number at least four. We also present a list of necessary conditions for a partition to have Schaper number at least three for odd primes and a conjecture on the sufficiency of these conditions.

Transitioning Between Tableaux and Spider Bases for Specht Modules

Regarding the Specht modules associated to the two-row partition (n, n), we provide a combinatorial path model to study the transitioning matrix from the tableau basis to the A1-web basis (i.e. cup diagrams), and prove that the entries in this matrix are positive in the upper-triangular portion with respect to a certain partial order.

On a generalization of Lie(k): A CataLAnKe theorem

We initiate a study of the representation of the symmetric group on the multilinear component of an n-ary generalization of the free Lie algebra, which we call a free LAnKe. Our central result is that the representation of the symmetric group S2n−1 on the multilinear component of the free LAnKe with 2n−1 generators is given by an irreducible representation whose dimension is the nth Catalan number. This leads to a more general result on eigenspaces of a certain linear operator, which has additional consequences. We also obtain a new presentation of Specht modules of staircase shape as a consequence of our central result.

On some trivial source Specht modules

The paper presented here focuses on the classification of trivial source Specht modules. We completely classify the trivial source Specht modules labelled by hook partitions. We also classify the trivial source Specht modules labelled by two-part partitions in the odd characteristic case. Moreover, in the even characteristic case, we prove a result for the classification of the trivial source Specht modules labelled by partitions with 2-weight 2, which justifies a conjecture of [16].

A Proof of the Murnaghan–Nakayama Rule Using Specht Modules and Tableau Combinatorics

The Murnaghan–Nakayama rule is a combinatorial rule for the character values of symmetric groups. We give a new combinatorial proof by explicitly finding the trace of the representing matrices in the standard basis of Specht modules. This gives an essentially bijective proof of the rule. A key lemma is an extension of a straightening result proved by the second author to skew tableaux. Our module theoretic methods also give short proofs of Pieri’s rule and Young’s rule.

Decomposable Specht modules indexed by bihooks

We study the decomposability of Specht modules labelled by bihooks, bipartitions with a hook in each component, for the Iwahori--Hecke algebra of type $B$. In all characteristics, we determine a large family of decomposable Specht modules, and conjecture that these provide a complete list of decomposable Specht modules indexed by bihooks. We prove the conjecture for small $n$.

Decompositions of some Specht modules I

We give a decomposition, as a direct sum of indecomposable modules, of several types of Specht modules in characteristic 2. These include the Specht modules labelled by hooks, whose decomposability was considered by Murphy, [15]. Since the main arguments are essentially no more difficult for Hecke algebras at parameter q=−1, we proceed in this generality.

Specht modules decompose as alternating sums of restrictions of Schur modules

Schur modules give the irreducible polynomial representations of the general linear group G L t \mathrm {GL}_t . Viewing the symmetric group S t \mathfrak {S}_t as a subgroup of G L t \mathrm {GL}_t , we may restrict Schur modules to S t \mathfrak {S}_t and decompose the result into a direct sum of Specht modules, the irreducible representations of S t \mathfrak {S}_t . We give an equivariant Möbius inversion formula that we use to invert this expansion in the representation ring for S t \mathfrak {S}_t for t t large. In addition to explicit formulas in terms of plethysms, we show the coefficients that appear alternate in sign by degree. In particular, this allows us to define a new basis of symmetric functions whose structure constants are stable Kronecker coefficients and which expand with signs alternating by degree into the Schur basis.

On the classification of Specht modules with one-dimensional summands

On the classification of Specht modules with one-dimensional summands

PI-eigenfunctions of the Star graphs

We consider the symmetric group Symn, n⩾2, generated by the set S of transpositions (1i),2⩽i⩽n, and the Cayley graph Sn=Cay(Symn,S) called the Star graph. For any positive integers n⩾3 and m with n>2m, we present a family of eigenfunctions of Sn with eigenvalue n−m−1 and call them PI-eigenfunctions. We then establish a connection of these functions with the standard basis of a Specht module. Finally, in the case of the largest non-principal eigenvalue n−2 we prove that any eigenfunction of Sn can be reconstructed by its values on the second neighbourhood of a vertex.

Specht modules labelled by hook bipartitions II

We continue the study of Specht modules labelled by hook bipartitions for the Iwahori–Hecke algebra of type B with e∈{3,4,…} via the cyclotomic Khovanov–Lauda–Rouquier algebra HnΛ. Over an arbitrary field, we explicitly determine the graded decomposition submatrices for HnΛ comprising rows corresponding to hook bipartitions.

Specht Modules for Quiver Hecke Algebras of Type $C$

We construct and investigate Specht modules S^{\lambda} for cyclotomic quiver Hecke algebras of types C^{(1)}_\ell and C_\infty , which are labelled by multipartitions \lambda . It is shown that in type C_\infty , the Specht module S^{\lambda} has a homogeneous basis indexed by standard tableaux of shape \lambda , which yields a graded character formula and good properties with the exact functors E_i^\Lambda and F_i^\Lambda . For type C^{(1)}_\ell , we propose a conjecture.

Restricting cell modules of partition algebras

The restriction of a (dual) Specht module to a smaller symmetric group has a filtration by (dual) Specht modules of this smaller group. In the cellular structure of the group algebra of the symmetric group, the cell modules are exactly the (dual) Specht modules. The partition algebra is a cellular algebra containing the group algebra of the symmetric group. In this article, we study the structure of the restriction of a cell module to the group algebra of a symmetric group (with smaller index) and find conditions for the restriction to possess a (dual) Specht filtration.

On C-Bases, Partition Pairs and Filtrations for Induced or Restricted Specht Modules

We obtain alternative explicit Specht filtrations for the induced and the restricted Specht modules in the Hecke algebra of the symmetric group (defined over the ring A = ℤ[q1/2, q−1/2], where q is an indeterminate) using C-bases for these modules. Moreover, we provide a link between a certain C-basis for the induced Specht module and the notion of pairs of partitions.