Shouchuan Zhang, Peng Wang, Jing Cheng, Hui Yang
To classify the finite dimensional pointed Hopf algebras with Weyl group $G$ of $E_6$, $E_7$, $F_4$, $G_2$, we obtain the representatives of conjugacy classes of $G$ and all character tables of centralizers of these representatives by means of software GAP.
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To classify the finite dimensional pointed Hopf algebras with $G= {\rm McL}$ we obtain the representatives of conjugacy classes of $G$ and all character tables of centralizers of these representatives by means of software {\rm GAP}.
To classify the finite dimensional pointed Hopf algebras with Weyl group $G$ of $E_8$, we obtain the representatives of conjugacy classes of $G$ and all character tables of centralizers of these representatives by means of software GAP. In this paper we only list character table 95--112.
To classify the finite dimensional pointed Hopf algebras with Weyl group $G$ of $E_8$, we obtain the representatives of conjugacy classes of $G$ and all character tables of centralizers of these representatives by means of software GAP. In this paper we only list character table 65--94.
To classify the finite dimensional pointed Hopf algebras with Weyl group $G$ of $E_8$, we obtain the representatives of conjugacy classes of $G$ and all character tables of centralizers of these representatives by means of software GAP. In this paper we only list character table 47--64.
To classify the finite dimensional pointed Hopf algebras with Weyl group $G$ of $E_8$, we obtain the representatives of conjugacy classes of $G$ and all character tables of centralizers of these representatives by means of software GAP. In this paper we only list character table 29--46.
To classify the finite dimensional pointed Hopf algebras with Weyl group $G$ of $E_8$, we obtain the representatives of conjugacy classes of $G$ and all character tables of centralizers of these representatives by means of software GAP. In this paper we only list character table 1-- 28.
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…
We analyze the centralizer of the Macdonald difference operator in an appropriate algebra of Weyl group invariant difference operators. We show that it coincides with Cherednik's commuting algebra of difference operators via an analog of the Harish-Chandra isomorphism. Analogs of Harish-Chandra series are defined and realized as solutions to the system of basic hypergeometric difference equations associated to the centralizer algebra. These Harish-Chandra series are then related to both Macdonald polynomials and Chalykh's Baker-Akhiezer functions.
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…
The centralizer of the image of the diagonal embedding of $U_q(\mathfrak{sl}_2)$ in the tensor product of three irreducible representations is examined in a Schur-Weyl duality spirit. The aim is to offer a description in terms of generators and relations. A conjecture in this respect is offered with the centralizers presented as quotients of the Askey-Wilson algebra. Support for the conjecture is provided by an examination of the representations of the quotients. The conjecture is also shown to be true in a number of cases thereby exhibiting in particular the Temperley-Lieb, Birman-Murakami-We…
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。
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.
Temperley-Lieb, Brauer and Racah algebras and other centralizers of su(2)
本文研究 Racah 代数与 su(2) 三个不可约表示张量积中心化子的关系，证明 Racah 代数满射到中心化子，并提出核的描述猜想，在若干情形下证明该猜想并得到 Temperley-Lieb、Brauer 等代数作为其商。
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 $…
We give a combinatorial description of a new diagram algebra, the partial Temperley--Lieb algebra, arising as the generic centralizer algebra $\mathrm{End}_{\mathbf{U}_q(\mathfrak{gl}_2)}(V^{\otimes k})$, where $V = V(0) \oplus V(1)$ is the direct sum of the trivial and natural module for the quantized enveloping algebra $\mathbf{U}_q(\mathfrak{gl}_2)$. It is a proper subalgebra of the Motzkin algebra (the $\mathbf{U}_q(\mathfrak{sl}_2)$-centralizer) of Benkart and Halverson. We prove a version of Schur--Weyl duality for the new algebras, and describe their generic representation theory.
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…
We prove that semi-infinite Bruhat order on an affine Weyl group is completely determined from those on the quotients by affine Weyl subgroups associated with various maximal (standard) parabolic subgroups of finite type. Furthermore, for an affine Weyl group of classical type, we give a complete classification of all cover relations of semi-infinite Bruhat order (or equivalently, all edges of the quantum Bruhat graphs) on the quotients in terms of tableaux. Combining these we obtain a tableau criterion for semi-infinite Bruhat order on an affine Weyl group of classical type. As an application…
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。
We give a new type of Schur-Weyl duality for the representations of a family of quantum subgroups and their centralizer algebra. We define and classify singly-generated, Yang-Baxter relation planar algebras. We present the skein theoretic construction of a new parameterized planar algebra. We construct infinitely many new subfactors and unitary fusion categories, and compute their trace formula as a closed-form expression, in terms of Young diagrams.
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。
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.
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。
We analyze the BGG Category $\mathcal{O}$ over a large class of generalized Weyl algebras (henceforth termed GWAs). Given such a "triangular" GWA for which Category $\mathcal{O}$ decomposes into a direct sum of subcategories, we study in detail the homological properties of blocks with finitely many simples. As consequences, we show that the endomorphism algebra of a projective generator of such a block is quasi-hereditary, finite-dimensional, and graded Koszul. We also classify all tilting modules in the block, as well as all submodules of all projective and tilting modules. Finally, we prese…
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…
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。
The affine and degenerate affine Birman-Murakami-Wenzl (BMW) algebras arise naturally in the context of Schur-Weyl duality for orthogonal and symplectic quantum groups and Lie algebras, respectively. Cyclotomic BMW algebras, affine and cyclotomic Hecke algebras, and their degenerate versions are quotients. In this paper we explain how the affine and degenerate affine BMW algebras are tantalizers (tensor power centralizer algebras) by defining actions of the affine braid group and the degenerate affine braid algebra on tensor space and showing that, in important cases, these actions induce acti…
Given a module $M$ for the algebra $\mathcal{D}_{\mathtt{q}}(G)$ of quantum differential operators on $G$, and a positive integer $n$, we may equip the space $F_n^G(M)$ of invariant tensors in $V^{\otimes n}\otimes M$, with an action of the double affine Hecke algebra of type $A_{n-1}$. Here $G= SL_N$ or $GL_N$, and $V$ is the $N$-dimensional defining representation of $G$. In this paper we take $M$ to be the basic $\mathcal{D}_{\mathtt{q}}(G)$-module, i.e. the quantized coordinate algebra $M= \mathcal{O}_{\mathtt{q}}(G)$. We describe a weight basis for $F_n^G(\mathcal{O}_{\mathtt{q}}(G))$ com…
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。
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