Howe Duality and Dynamical Weyl Group.

Abstract

We give a fermionic formula for R-matrices of exterior powers of the vector representations of $U_q\left({\widehat{\mathrm{gl}}}_N\right)$ and relate it to the dynamical Weyl group of Tarasov-Varchenko and Etingof-Varchenko, via a Howe ( ${\mathrm{gl}}_N,{\mathrm{gl}}_M)$ -duality. In the limit $N\to \infty$ we obtain R-matrices for Fock spaces. As a consequence of our result we obtain a dynamical action of the Weyl group on integrable $U_q{\mathrm{gl}}_M$ -modules, extending the known action on zero weight spaces. In an Appendix by Anfisa Gurenkova it is shown that the latter property also holds if we replace ${\mathrm{gl}}_M$ by a general symmetrizable Kac-Moody algebra.