Schur-Weyl reciprocity for the q-analogue of the alternating group

Specht modules decompose as alternating sums of restrictions of Schur modules

Schur modules give the irreducible polynomial representations of the general linear group $\mathrm{GL}_t$. Viewing the symmetric group $\mathfrak{S}_t$ as a subgroup of $\mathrm{GL}_t$, we may restrict Schur modules to $\mathfrak{S}_t$ and decompose the result into a direct sum of Specht modules, the irreducible representations of $\mathfrak{S}_t$. We give an equivariant Möbius inversion formula that we use to invert this expansion in the representation ring for $\mathfrak{S}_t$ for $t$ large. In addition to explicit formulas in terms of plethysms, we show the coefficients that appear alternat…

Schur Algebras for the Alternating Group and Koszul Duality

We introduce the alternating Schur algebra $AS_F(n,d)$ as the commutant of the action of the alternating group $A_d$ on the $d$-fold tensor power of an $n$-dimensional $F$-vector space. When $F$ has characteristic different from $2$, we give a basis of $AS_F(n,d)$ in terms of bipartite graphs, and a graphical interpretation of the structure constants. We introduce the abstract Koszul duality functor on modules for the even part of any $\mathbf Z/2\mathbf Z$-graded algebra. The algebra $AS_F(n,d)$ is $\mathbf Z/2\mathbf Z$-graded, having the classical Schur algebra $S_F(n,d)$ as its even part. …

对称群双覆盖的RoCK块与广义Schur代数

RoCK blocks of double covers of symmetric groups and generalized Schur algebras

本文研究对称群与交错群双覆盖的块，通过广义Schur超代数给出任意亏数RoCK块的局部描述，并在导出等价下推广至任意块。

A super Frobenius formula for the characters of cyclotomic Hecke algebras

We prove a super Frobenius formula for the characters of the cyclotomic Hecke algebras by applying the super Schur-Weyl reciprocity between the quantum superalgebras and cyclotomic Hecke algebras, which is a super analogue of the Frobenius formula in (T. Shoji, J. Algebra 226: 2000, 818--856) and a cyclotomic analogue of the super Frobeinus formula in (H. Mitsuhashi, Linear Multilinear Algebra 58: 2010, 941--955).

Factorization theorems for classical group characters, with applications to alternating sign matrices and plane partitions

We show that, for a certain class of partitions and an even number of variables of which half are reciprocals of the other half, Schur polynomials can be factorized into products of odd and even orthogonal characters. We also obtain related factorizations involving sums of two Schur polynomials, and certain odd-sized sets of variables. Our results generalize the factorization identities proved by Ciucu and Krattenthaler (Advances in combinatorial mathematics, 39-59, 2009) for partitions of rectangular shape. We observe that if, in some of the results, the partitions are taken to have rectangul…

Irreducible projective representations of the alternating group which remain irreducible in characteristic 2

For any finite group G it is an interesting question to ask which ordinary irreducible representations of G remain irreducible in a given characteristic p. We answer this question for p=2 when G is the proper double cover of the alternating group. As a key ingredient in the proof, we prove a formula for the decomposition numbers in Rouquier blocks of double covers of symmetric groups, in terms of Schur P-functions.

对称群与交错群的旋表示不可约限制

Irreducible restrictions of spin representations of symmetric and alternating groups

本文研究代数闭域上几乎拟单群中，限制不可约的子群与模的分类问题，并聚焦于交错群与对称群的 Schur 双覆盖情形。

Dimensions of irreducible modules for partition algebras and tensor power multiplicities for symmetric and alternating groups

The partition algebra $\mathsf{P}_k(n)$ and the symmetric group $\mathsf{S}_n$ are in Schur-Weyl duality on the $k$-fold tensor power $\mathsf{M}_n^{\otimes k}$ of the permutation module $\mathsf{M}_n$ of $\mathsf{S}_n$, so there is a surjection $\mathsf{P}_k(n) \to \mathsf{Z}_k(n) := \mathsf{End}_{\mathsf{S}_n}(\mathsf{M}_n^{\otimes k}),$ which is an isomorphism when $n \ge 2k$. We prove a dimension formula for the irreducible modules of the centralizer algebra $\mathsf{Z}_k(n)$ in terms of Stirling numbers of the second kind. Via Schur-Weyl duality, these dimensions equal the multiplicities …

一个新的对称函数恒等式及其在对称群特征值中的应用

A New Symmetric Function Identity With an Application to symmetric group character values

本文研究对称群特征和猜想，通过Schur函数与幂和对称函数内积分析Ev(λ)上的特征交错和，发现幂和对称多项式交错和与单项式对称多项式乘积的恒等式，证明了该特征交错和的一个特例为0。

箭图 Hecke 超代数的虚 Schur-Weyl 对偶

Imaginary Schur-Weyl duality for quiver Hecke superalgebras

本文研究箭图 Hecke 超代数的虚尖代数，通过构造非负分次的 Morita 等价代数 C(n,d)，将其表示论归结为经典 Schur 代数，从而用多重分拆典范分类不可约模。

Spin representations and centralizer algebras for the Spinor groups

We pursue an analogy of the Schur-Weyl reciprocity for the spinor groups and pick up the irreducible spin representations in the tensor space $Δ\textstyle{\bigotimes \bigotimes^k V}$. Here $Δ$ is the fundamental representation of $Pin(N)$ and $V$ is the natural (vector) representation of the orthogonal group O(N). We consider the centralizer algebra $\mathbf{CP_k} = Pin(N)(Δ\textstyle{\bigotimes \bigotimes^k V})$ for $Pin(N)$, the double covering group of O(N) and define two kinds of linear basis in $\mathbf{CP_k}$ (one comes from invariant theory and the other from representation theory), bot…