Schur-Weyl dualities for symmetric inverse semigroups

Two dualities: Markov and Schur-Weyl

We show that quantum Schur-Weyl duality leads to Markov duality for a variety of asymmetric interacting particle systems. In particular, we consider three cases: (1) Using a Schur-Weyl duality between a two-parameter quantum group and a two-parameter Hecke algebra from arXiv:math/0108038, we recover the Markov self-duality of multi-species ASEP previously discovered in arXiv:1605.00691 and arXiv:1606.04587. (2) From a Schur-Weyl duality between a co-ideal subalgebra of a quantum group and a Hecke algebra of type B arXiv:1609.01766, we find a Markov duality for a multi-species open ASEP on the …

Schur-Weyl duality as an instrument of Gauge-String duality

A class of mathematical dualities have played a central role in mapping states in gauge theory to states in the spacetime string theory dual. This includes the classical Schur-Weyl duality between symmetric groups and Unitary groups, as well as generalisations involving Brauer and Hecke algebras. The physical string dualities involved include examples from the AdS/CFT correspondence as well as the string dual of two-dimensional Yang Mills.

Quasi-split symmetric pairs of $U(\mathfrak{gl}_N)$ and their Schur algebras

We establish explicit isomorphisms of two seemingly-different algebras, and their Schur algebras, arising from the centralizers of two different type B Weyl group actions in Schur-like dualities. We provide a presentation of the geometric counterpart of the above Schur algebras in Bao-Kujawa-Li-Wang (2018) specialized at $q=1$.

广义 Schur-Weyl 对偶性：量子仿射对称对与定向 KLR 代数

Generalized Schur-Weyl dualities for quantum affine symmetric pairs and orientifold KLR algebras

该文构造量子仿射对称对与定向 KLR 代数间的函子，推广广义 Schur-Weyl 对偶性，并恢复拟分裂 AIII 型的已知对偶结果。

On the affine Schur algebra of type A

The affine Schur algebra $\widetilde{S}(n,r)$ (of type A) over a field $K$ is defined to be the endomorphism algebra of the tensor space over the extended affine Weyl group of type $A_{r-1}$. By the affine Schur-Weyl duality it is isomorphic to the image of the representation map of the $\mathcal{U}(\hat{\mathfrak{gl}}_{n})$ action on the tensor space when $K$ is the field of complex numbers. We show that $\widetilde{S}(n,r)$ can be defined in another two equivalent ways. Namely, it is the image of the representation map of the semigroup algebra $K\widetilde{GL}_{n,a}$ (defined in Section \ref…

Infnite-dimensional Schur-Weyl duality and the Coxeter-Laplace operator

We extend the classical Schur-Weyl duality between representations of the groups $SL(n,\C)$ and $\sN$ to the case of $SL(n,\C)$ and the infinite symmetric group $\sinf$. Our construction is based on a "dynamic," or inductive, scheme of Schur-Weyl dualities. It leads to a new class of representations of the infinite symmetric group, which have not appeared earlier. We describe these representations and, in particular, find their spectral types with respect to the Gelfand-Tsetlin algebra. The main example of such a representation acts in an incomplete infinite tensor product. As an important app…

A categorical approach to classical and quantum Schur-Weyl duality

We use category theory to propose a unified approach to the Schur-Weyl dualities involving the general linear Lie algebras, their polynomial extensions and associated quantum deformations. We define multiplicative sequences of algebras exemplified by the sequence of group algebras of the symmetric groups and use them to introduce associated monoidal categories. Universal properties of these categories lead to uniform constructions of the Drinfeld functor connecting representation theories of the degenerate affine Hecke algebras and the Yangians and of its q-analogue. Moreover, we construct act…

Geometric Howe dualities of finite type

We develop a geometric approach toward an interplay between a pair of quantum Schur algebras of arbitrary finite type. Then by Beilinson-Lusztig-MacPherson's stabilization procedure in the setting of partial flag varieties of type A (resp. type B/C), the Howe duality between a pair of quantum general linear groups (resp. a pair of $\imath$quantum groups of type AIII/IV) is established. The Howe duality for quantum general linear groups has been provided via quantum coordinate algebras in [Z02]. We also generalize this algebraic approach to $\imath$quantum groups of type AIII/IV, and prove that…

Schur sector of Argyres-Douglas theory and $W$-algebra

We study the Schur index, the Zhu's $C_2$ algebra, and the Macdonald index of a four dimensional $\mathcal{N}=2$ Argyres-Douglas (AD) theories from the structure of the associated two dimensional $W$-algebra. The Schur index is derived from the vacuum character of the corresponding $W$-algebra and can be rewritten in a very simple form, which can be easily used to verify properties like level-rank dualities, collapsing levels, and S-duality conjectures. The Zhu's $C_2$ algebra can be regarded as a ring associated with the Schur sector, and a surprising connection between certain Zhu's $C_2$ al…

G(l,k,d)-modules via groupoids

In this note we describe a seemingly new approach to the complex representation theory of the wreath product $G\wr S_d$ where $G$ is a finite abelian group. The approach is motivated by an appropriate version of Schur-Weyl duality. We construct a combinatorially defined groupoid in which all endomorphism algebras are direct products of symmetric groups and prove that the groupoid algebra is isomorphic to the group algebra of $G\wr S_d$. This directly implies a classification of simple modules. As an application, we get a Gelfand model for $G\wr S_d$ from the classical involutive Gelfand model …

Rook 特征标作为对称函数

Rook characters as symmetric functions

本文刻画了对称函数环中一个非齐次基，其求值对应 rook 幺半群不可约表示特征标，并利用 Schur–Weyl 对偶揭示其与传播分拆代数及 Schur 函数 smash 积的关联。

Temperley-Lieb代数的拟遗传覆盖与相对控制维数

Quasi-hereditary covers of Temperley-Lieb algebras and relative dominant dimension

本文研究Temperley-Lieb代数的拟遗传覆盖问题，通过证明拟遗传代数相对控制维数的两倍关系，应用于Schur代数确定其相对控制维数，得到q-模拟及Hemmer-Nakano型结果，首次完整分类相关联系并确定整系数最佳拟遗传覆盖。

Schur-Weyl duality for tensor powers of the Burau representation

Artin's braid group $B_n$ is generated by $σ_1, \dots, σ_{n-1}$ subject to the relations \[ σ_i σ_{i+1} σ_i = σ_{i+1} σ_i σ_{i+1}, \quad σ_iσ_j = σ_j σ_i \text{ if } |i-j|>1. \] For complex parameters $q_1,q_2$ such that $q_1q_2 \ne 0$, the group $B_n$ acts on the vector space $\mathbf{E} = \sum_i \mathbb{C} \mathbf{e}_i$ with basis $\mathbf{e}_1, \dots, \mathbf{e}_n$ by \begin{gather*} σ_i \cdot \mathbf{e}_i = (q_1+q_2)\mathbf{e}_i + q_1\mathbf{e}_{i+1}, \quad σ_i \cdot \mathbf{e}_{i+1} = -q_2\mathbf{e}_i, \\ σ_i \cdot \mathbf{e}_j = q_1 \mathbf{e}_j \text{ if } j \ne i,i+1. \end{gather*} Thi…

Polynomial representations of $C^*$-algebras and their applications

This is a sequel to our paper on nonlinear completely positive maps and dilation theory for real involutive algebras, where we have reduced all representation classification problems to the passage from a $C^*$-algebra ${\mathcal A}$ to its symmetric powers $S^n({\mathcal A})$, resp., to holomorphic representations of the multiplicative $*$-semigroup $({\mathcal A},\cdot)$. Here we study the correspondence between representations of ${\mathcal A}$ and of $S^n({\mathcal A})$ in detail. As $S^n({\mathcal A})$ is the fixed point algebra for the natural action of the symmetric group $S_n$ on ${\ma…