Polynomial invariants are fundamental objects in analysis on Lie groups and symmetric spaces. Invariant differential operators on symmetric spaces are described by Weyl group invariant polynomial. In this article we give a simple criterion that ensure that the restriction of invariant polynomials to subspaces is surjective. In another paper we will apply our criterion to problems in Fourier analysis on projective/injective limits, specifically to theorems of Paley--Wiener type.
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, sta…
We prove Turner's conjecture, which describes the blocks of the Hecke algebras of the symmetric groups up to derived equivalence as certain explicit Turner double algebras. Turner doubles are Schur-algebra-like `local' objects, which replace wreath products of Brauer tree algebras in the context of the Broué abelian defect group conjecture for blocks of symmetric groups with non-abelian defect groups. The main tools used in the proof are generalized Schur algebras corresponding to wreath products of zigzag algebras and imaginary semicuspidal quotients of affine KLR algebras.
On a relation between the basic representation of the affine Lie algebra $\widehat\sl$ and a Schur--Weyl representation of the infinite symmetric group
Let $K$ be an algebraically closed field of characteristic $p\geqslant 0$ and let $W$ be a finite-dimensional $K$-space of dimension greater than or equal to $5.$ In this paper, we give the structure of certain Weyl modules for $G=\mbox{Spin}(W)$ in the case where $p\neq 2,$ as well as the dimension of the corresponding irreducible, finite-dimensional, rational $KG$-modules. In addition, we determine the composition factors of the restriction of certain irreducible, finite-dimensional, rational $K\mbox{SL}(W)$-modules to $\mbox{SO}(W).$
We define a graded graph, called the Schur--Weyl graph, which arises naturally when one considers simultaneously the RSK algorithm and the classical duality between representations of the symmetric and general linear groups. As one of the first applications of this graph, we give a new proof of the completeness of the list of discrete indecomposable characters of the infinite symmetric group.
In this paper, we give the decomposition into irreducible characters of the restriction to the wreath product $\mathbb{Z}_{p-1} \wr \mathfrak{S}_w$ of any irreducible character of $(\mathbb{Z}_p \rtimes \mathbb{Z}_{p-1}) \wr \mathfrak{S}_w$, where $p$ is any odd prime, $w \geq 0$ is an integer, and $\mathbb{Z}_p$ and $\mathbb{Z}_{p-1}$ denote the cyclic groups of order $p$ and $p-1$ respectively. This answers the question of how to decompose the restrictions to $p$-regular elements of irreducible characters of the symmetric group $\mathfrak{S}_n$ in the $\mathbb{Z}$-basis corresponding to the …
The so-called Garnir relations are very important in the representation theory of the symmetric groups. In this paper, we give a possible extension of Garnir relations for Weyl groups in general.
Two super-analogs of the Schur-Weyl duality are considered: the duality of actions in $(\mathbb{C}^{m|n})^{\otimes N}$ of the Lie superalgebra $\mathfrak{gl}(m,n)$ and the symmetric group $S_N$, and the duality of actions of the Lie superalgebra $Q(n)$ and a certain finite group $Se(N)$ in $(\mathbb{C}^{n|n})^{\otimes N}$. We construct an isomorphism of symmetric and universal enveloping algebras of these Lie superalgebras called special symmetrization. Using this isomorphism of vector spaces we describe explicitly the duality between the centers of the corresponding universal enveloping algeb…
Let $(G,G^Γ)$ be a Klein four symmetric pair. The author wants to classify all the Klein four symmetric pairs $(G,G^Γ)$ such that there exists at least one nontrivial unitarizable simple $(\mathfrak{g},K)$-module $π_K$ that is discretely decomposable as a $(\mathfrak{g}^Γ,K^Γ)$-module. In this article, three assumptions will be made. Firstly, $G$ is an exceptional Lie group of Hermitian type, i.e., $G=\mathrm{E}_{6(-14)}$ or $\mathrm{E}_{7(-25)}$. Secondly, $G^Γ$ is noncompact. Thirdly, there exists an element $σ\inΓ$ corresponding to a symmetric pair of anti-holomorphic type such that $π_K$ i…
Let $\mathbb{C}\mathsf{A}_n = \mathbb{C}[S_2\wr S_2 \wr\cdots \wr S_2]$ be the group algebra of $n$-step iterated wreath product. We prove some structural properties of $\mathsf{A}_n$ such as their centers, centralizers, right and double cosets. We apply these results to explicitly write down Mackey theorem for groups $\mathsf{A}_n$ and give a partial description of the natural transformations between induction and restriction functors on the representations of the iterated wreath product tower by computing certain hom spaces of the category of $\displaystyle \bigoplus_{m\geq 0}(\mathsf{A}_m, …
A family of symmetric functions $\tilde{s}_λ$ was introduced in [OZ], and independently in [AS]. The $\tilde{s}_λ$ encode many stability properties of representations of symmetric groups (e.g. when multiplied, the structure constants are reduced Kronecker coefficients). We show that the structure constants for the Kronecker comultiplication $Δ^*$ are multiplicities for the restriction of irreducible representations from $S_{mn}$ to $S_m \times S_n$ (provided $m$ and $n$ are sufficiently large), and use the structure of $\tilde{s}_λ$ to demonstrate two-row stability properties of these restrict…
Young's orthogonal basis is a classical basis for an irreducible representation of a symmetric group. This basis happens to be a Gelfand-Tsetlin basis for the chain of symmetric groups. It is well-known that the chain of alternating groups, just like the chain of symmetric groups, has multiplicity-free restrictions for irreducible representations. Therefore each irreducible representation of an alternating group also admits Gelfand-Tsetlin bases. Moreover, each such representation is either the restriction of, or a subrepresentation of, the restriction of an irreducible representation of a sym…
Building on reduction theorems and dimension bounds for symmetric groups obtained in our earlier work, we classify the irreducible restrictions of representations of the symmetric and alternating groups to proper subgroups. Such classification is known when the characteristic of the ground field is greater than $3$, but the small characteristics cases require a substantially more delicate analysis and new ideas. Our results fit into the Aschbacher-Scott program on maximal subgroups of finite classical groups.
We construct the polynomial induction functor, which is the right adjoint to the restriction functor from the category of polynomial representations of a general linear group to the category of representations of its Weyl group. This construction leads to a representation-theoretic proof of Littlewood's plethystic formula for the multiplicity of an irreducible representation of the symmetric group in such a restriction. The unimodality of certain bipartite partition functions follows.
We study irreducible restrictions from modules over symmetric groups to subgroups. We get reduction results which substantially restrict the classes of subgroups and modules for which this is possible. Such results are known when the characteristic of the ground field is greater than $3$, but the small characteristics cases require a substantially more delicate analysis and new ideas. This work fits into the Aschbacher-Scott program on maximal subgroups of finite classical groups.
This note presents a procedure to determine the reduction of the irreducible and the induced characters of the symmetric group in terms of the irreducible and induced characters of the hyperoctahedral group Key Words: Symmetric Group, Hyperoctahedral group, Representations, Characters, Re- duction.
We relate properties of weighted flags (or multiflags) of type AD to statistics of the corresponding Weyl groups. For type A, we recover the Mahonian statistics on symmetric groups. Finally, we sketch briefly an easy extension incorporating statistics for so-called Euler-polynomials.
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.
