Quantized mixed tensor space and Schur-Weyl duality

Abstract

Let $R$ be a commutative ring with one and $q$ an invertible element of $R$. The (specialized) quantum group $\mathbf{U}=U_q({\mathfrak{gl}}_n)$ over $R$ of the general linear group acts on mixed tensor space $V^{\otimes r}\otimes {V^{\ast }}^{\otimes s}$ where $V$ denotes the natural $\mathbf{U}$-module $R^n$, $r,s$ are nonnegative integers and $V^{\ast }$ is the dual $\mathbf{U}$-module to $V$. The image of $\mathbf{U}$ in ${\mathrm{End}}_R(V^{\otimes r}\otimes {V^{\ast }}^{\otimes s})$ is called the rational $q$-Schur algebra $S_q(n;r,s)$. We construct a bideterminant basis of $S_q(n;r,s)$. There is an action of a $q$-deformation ${\mathfrak{B}}_{r,s}^n(q)$ of the walled Brauer algebra on mixed tensor space centralizing the action of $\mathbf{U}$. We show that ${\mathrm{End}}_{{\mathfrak{B}}_{r,s}^n(q)}(V^{\otimes r}\otimes {V^{\ast }}^{\otimes s})=S_q(n;r,s)$. By \cite{dipperdotystoll} the image of ${\mathfrak{B}}_{r,s}^n(q)$ in ${\mathrm{End}}_R(V^{\otimes r}\otimes {V^{\ast }}^{\otimes s})$ is ${\mathrm{End}}_{\mathbf{U}}(V^{\otimes r}\otimes {V^{\ast }}^{\otimes s})$. Thus mixed tensor space as $\mathbf{U}$-${\mathfrak{B}}_{r,s}^n(q)$-bimodule satisfies Schur-Weyl duality.

31 pages

References in corpus (1)

• Schur-Weyl duality for orthogonal groups

Cited by in corpus (2)

• The representations of quantized walled Brauer algebras
• Decomposition numbers of quantized walled Brauer algebras