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3:30 p.m. - 4:45 p.m. Deformations of $U_q(gl_{mn})$ modules - an approach to the Kronecker problem K V Subrahmanyam Chennai Mathematical Institute. 18-04-13 Abstract Let G=GL(mn) and $\lambda$ a partition of an integer $d$ into at most $mn$ parts. Let V_{\lambda}(mn) be the irreducible GL(mn) module corresponding to shape $\lambda$. Consider the Kronecker embedding, $GL(m) \otimes GL(n) in $GL(mn)$ given by $(A,B) \rightarrow (A \otimes B)$. Given partitions $\mu$, $\nu$ of $d$ into at most $m$ and $n$ parts respectivey, the Kronecker problem is to determine the multiplicities of the irreducible $GL(m) \times GL(n)$ module $V_{\mu}(m) \otimes V_{\nu}(n)$ in $V_{\lambda}(mn)$. We present an approach to this problem using Quantum groups. In the first part of the talk we consider the case when $\lambda$ is miniscule i.e. $V_{\lambda}(mn)$ is the exterior power of the standard representation $C^{mn}. The solution to the Kronecker problem in this case is well known. However we propose a different proof based on quantum groups. We show that there is a $U_q(gl_m) \otimes U_q(gl_n)$-commuting action on the quantum space $\wedge^d_q(mn)$. A consequence of this is that there is a Kashiwara theory of crystals and crystal operators, which allows us to solve the Kronecker problem. We show that signed subsets form a crystal basis. Our $U_q(gl_m)$ and $U_q(gl_n)$ crystal operators on the crystal basis will be exactly what were discovered by Koshevoi and Vinberg in their combinatorial approach to this problem. So we have a quantum explanation of their combinatorial rules. In the next part of the talk, we will see how to use the commuting action on miniscule representations, to make one more step towards the Kronecker problem. We will show that in the case when $n=2$, for every $\lambda$ a partition of an integer $d$ into at most 2m parts, there is a vector space $W_{q,\lambda}(2m)$ with a commuting $U_q(gl_2) \otimes U_q(gl_m)$ action, such that at $q=1$ the action of $U(gl_m) \otimes U(gl_2)$ on $W_{1,\lambda}(2m)$ coincides with the $U(gl_m) \otimes U(gl_2)$ action on $V_{\lambda}(2m)$ coming from the Kronecker embedding. So we have a deformation of the irreducible $GL(2m)$ module of shape $\lambda$ with a commuting bi-action. From this one hopes to discover a crystal basis and crystal operators on the crystal basis to solve the Kronecker problem. This is joint work with Bharat Adsul and Milind Sohoni from IIT Bombay.
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