The character tables of centralizers in Weyl Group of $E_8$ III

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

We establish Schur-Weyl reciprocity for the q-analogue of the alternating group. We analyze the sign q-permutation representation of the Hecke algebra on the tensor product of the super vector space V in detail, and examine its restriction to the q-analogue of the alternating group. In consequence, we find out that if the dimensions of even part and odd part of V are same, then the centralizer of the q-analogue of the alternating group is a Z_2-crossed product of the centralizer of the Hecke algebra and obtain Schur-Weyl reciprocity between them. Though the structure of the centralizer is more…

Traces on diagram algebras II: Centralizer algebras of easy groups and new variations of the Young graph

In continuation of our recent work arXiv:2006.07312, we classify the extremal traces on infinite diagram algebras that appear in the context of Schur-Weyl duality for Banica and Speicher's easy groups. We show that the branching graphs of these algebras describe walks on new variations of the Young graph which describe curious ways of growing Young diagrams. As a consequence, we prove that the extremal traces on generic rook-Brauer algebras are always extensions of extremal traces on the group algebra $\mathbb{C}[S_{\infty}]$ of the infinite symmetric group. Moreover, we conjecture that the sa…

量子化Weyl代数、双重中心化性质及$U_q(\mathfrak{gl}_n)$的一个新的第一基本定理

Quantized Weyl algebras, the double centralizer property, and a new First Fundamental Theorem for $U_q(\mathfrak{gl}_n)$

本文定义了矩阵空间量子化坐标环上的多项式系数微分算子代数的$q$-模拟，证明两个量子包络代数像与其交集互为互中心化子，从而建立了$U_q(\mathfrak{gl}_n)$不变理论的新第一基本定理，并给出Cartan子代数像交集的$q$-行列式生成元公式。

Representations of Two-Parameter Quantum Groups and Schur-Weyl Duality

We determine the finite-dimensional simple modules for two-parameter quantum groups corresponding to the general linear and special linear Lie algebras gl_n and sl_n, and give a complete reducibility result. These quantum groups have a natural n-dimensional module V. We prove an analogue of Schur-Weyl duality in this setting: the centralizer algebra of the quantum group action on the k-fold tensor power of V is a quotient of a Hecke algebra for all n and is isomorphic to the Hecke algebra in case n\geq k.

Higher frobenius-schur indicators for drinfeld doubles of finite groups through characters of centralizers

We present a new approach to calculating the higher Frobenius-Schur indicators for the simple modules over the Drinfeld double of a finite group. In contrast to the formula by Kashina-Sommerh{ä}user-Zhu that involves a sum over all group elements satisfying a certain condition, our formula operates on the level of conjugacy classes and character tables. It can be implemented in the computer algebra system GAP, efficiently enough to deal, on a laptop, with symmetric groups up to $S_{18}$ (providing further evidence that indicators are non-negative in this case) or simple groups of order up to $…

Frobenius-Schur Indicators for Subgroups and the Drinfel'd Double of Weyl Groups

If G is a finite group and k is a field, there is a natural construction of a Hopf algebra over k associated to G, the Drinfel'd double D(G). We prove that if G is any finite real reflection group with Drinfel'd double D(G) over an algebraically closed field k of characteristic not 2, then every simple D(G)-module has Frobenius-Schur indicator +1. This generalizes the classical results for modules over the group itself. We also prove some new results about Weyl groups. In particular, we prove that any abelian subgroup is inverted by some involution. Also, if E is any elementary abelian 2-subgr…

B型Duplex Hecke代数

Duplex Hecke Algebras of type B

本文引入B型duplex Hecke代数，推广了A型相关代数，证明其在张量空间有自然表示，并建立B型Levi型q-Schur-Weyl对偶，揭示该代数与ı量子群的双中心化子性质。

偏微分Hecke代数、Schur-Weyl对偶性与Frobenius特征公式

Mirabolic Hecke algebras, Schur-Weyl duality and Frobenius character formulas

本文研究了偏微分Hecke代数的新的呈现方式，建立了其与量子群之间的Schur-Weyl对偶性，并推导出Frobenius特征公式及递归的Murnaghan-Nakayama规则。

Explicit irreducible representations of the Iwahori-Hecke algebra of type $F_4$

A general method for computing irreducible representations of Weyl groups and Iwahori-Hecke algebras was introduced earlier by the first author. The representations of the algebras of types $A_n$, $B_n$, $D_n$ and $G_2$ were computed and it is the purpose of this paper to extend these computations to $F_4$. The main goal here is to compute irreducible representations of the Iwahori-Hecke algebra of type $F_4$ by only using information in the character table of the Weyl group.

mirabolic Hecke代数的量子超Schur–Weyl对偶与特征标公式

Quantum super Schur--Weyl duality and character formulas for mirabolic Hecke algebras

该研究建立了mirabolic Hecke代数的量子超Schur–Weyl对偶，推导了其不可约特征标的相关公式，还构造了超mirabolic RSK双射，并建立了其与q-rook幺半群代数的参数反转同构以转移相关结果。

The Heisenberg double of the quantum Euclidean group and its representations

The Heisenberg double $D_q(E_2)$ of the quantum Euclidean group $\mathcal{O}_q(E_2)$ is the smash product of $\mathcal{O}_q(E_2)$ with its Hopf dual $U_q(\mathfrak{e}_2)$. For the algebra $D_q(E_2)$, explicit descriptions of its prime, primitive, and maximal spectra are obtained. All prime factors of $D_q(E_2)$ are presented as generalized Weyl algebras. As a result, we obtain that the algebra $D_q(E_2)$ has no finite-dimensional representations, and that $D_q(E_2)$ cannot have a Hopf algebra structure. The automorphism groups of the quantum Euclidean group and its Heisenberg double are determ…

正特征下具有有限算术根系的对角型高阶Nichols代数

Higher rank Nichols algebras of diagonal type with finite arithmetic root system in positive characteristic

本文分类了正特征域上秩r≥5且具有有限不可约根系的对角型Nichols代数，通过Weyl群胚、算术根系和Cartan图证明其对应于表1中的广义Dynkin图。

关于互反特征与量子仿射Schur-Weyl对偶

On reciprocal characters and the quantum affine Schur-Weyl duality

本文通过仿射Hecke代数的Langlands/Zelevinsky参数化识别了Frenkel-Reshetikhin q-特征的主导部分，并引入互反特征，在量子仿射Schur-Weyl对偶下证明其与量子仿射代数有限维模的主导q-特征匹配，实现了Nakajima关于主导单项式在Lusztig框架中作用的A型实现。