3.45 p.m.
Towards the construction of quantum deformations of restrictions of GL(mn) module to GL(m) X GL(n).
K.V. Subrahmanyam
Chennai Mathematical Institute.
02-09-09

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Abstract

We would like to understand the decomposition of a GL(mn) irreducible module when restricted to GL(m) X GL(n). In order to do this, we wish to construct for each $U_q(gl (mn))$ irreducible module $V^q_{\lambda}$ a $U_q(gl(m)) \otimes U_q(gl(n))$ module $W_{\lambda}$, with certain nice properties - in particular we would like that, at q=1, the restriction of $V^1_{\lambda}$ to $U_1(gl(m)) \otimes U_1(gl(n))$ coincides with the action of $U_1(gl(m)) \otimes U_1(gl(n))$ on $W_{\lambda}$ at q=1.

We construct such a deformation $W$ when $V$ is the wedge representation of GL(mn), and when $V$ is $Sym^d(mn)$. We also describe some straightening laws, which seem to extend this construction when $\lambda$ has two columns.

We describe a $U_q(gl(m)) \otimes U_q(gl(n))$ crystals basis for wedge representations and a crystal basis for $Sym^d(mn)$.

This is joint work with Milind Sohoni and Bharat Adsul.