Howe duality and dynamical Weyl group

We give a fermionic formula for $R$-matrices of exterior powers of the vector representations of $U_q({\widehat{\mathfrak{g}\mathfrak{l}}}_N)$ and relate it to the dynamical Weyl group of Tarasov–Varchenko and Etingof–Varchenko, via a Howe (${\mathfrak{g}\mathfrak{l}}_N,{\mathfrak{g}\mathfrak{l}}_M)$-duality. In the limit $N\to \mathrm{\infty }$ we obtain $R$-matrices for Fock spaces. As a consequence of our result we obtain a dynamical action of the Weyl group on integrable $U_q{\mathfrak{g}\mathfrak{l}}_M$-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 ${\mathfrak{g}\mathfrak{l}}_M$ by a general symmetrizable Kac–Moody algebra.