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

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 $V_λ (\C^{mn})$ of $U_q(g{l}_{mn})$ we construct a representation $W_{\lambda }$ of $U_q(g{l}_m)\otimes U_q(g{l}_n)$ such that at $q=1$, the restriction of $V_λ (\C^{mn})$ to $U_1(g{l}_m)\otimes U_1(g{l}_n)$ 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 $U_q(g{l}_m)\otimes U_q(g{l}_n)$-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 $U_q(g{l}_m)\otimes U_q(g{l}_n)$-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 $W_{\lambda }$.

28 pages

References in corpus (1)

• Geometric Complexity Theory IV: nonstandard quantum group for the Kronecker problem