Quantum deformations of the restriction of $GL_{mn}(\C)$-modules to $GL_m(\C) \times GL_n(\C)$

TL;DR

This paper constructs a q-deformation of modules obtained by restricting $GL_{mn}( ext{C})$-modules to $GL_m( ext{C}) imes GL_n( ext{C})$, providing explicit modules and bases that generalize classical representations.

Contribution

It introduces a new construction of q-deformed modules for the restriction of $GL_{mn}( ext{C})$-modules to $GL_m( ext{C}) imes GL_n( ext{C})$, including bi-crystal bases and equivariant maps.

Findings

Constructed $U_q(gl_m) imes U_q(gl_n)$-modules $igwedge^k$ as q-deformations.

Developed bi-crystal bases consisting of signed subsets.

Established equivariant maps for module construction.

Abstract

In this paper, we consider the restriction of finite dimensional $GL_{mn} (\C)$-modules to the subgroup ${GL_m (\C)\times GL_n (\C)}$. In particular, for a Weyl module $V_{\lambda} (\C^{mn})$ of $U_q(gl_{mn})$ we construct a representation $W_{\lambda}$ of $U_q (gl_m)\otimes U_q (gl_n)$ such that at $q=1$, the restriction of $V_{\lambda} (\C^{mn})$ to $U_1 (gl_m)\otimes U_1 (gl_n)$ matches its action on $W_{\lambda}$ at $q=1$. Thus $W_{\lambda}$ is a $q$-deformation of the module $V_{\lambda}$. This is achieved by first constructing a $U_q (gl_m)\otimes U_q (gl_n)$-module $\wedge^k $, a $q$-deformation of the simple $GL_{mn} (\C)$-module $\wedge^k (\C^{mn})$. We also construct the bi-crystal basis for $\wedge^k $ and show that it consists of signed subsets. Next, we develop $U_q (gl_m) \otimes U_q (gl_n)$-equivariant maps $\psi_{a,b} :\wedge^{a+1} \otimes \wedge^{b-1} \to \wedge^a \otimes \wedge^b$. This is used as the building block to construct the general $W_{\lambda}$.