Howe duality and dynamical Weyl group

We give a fermionic formula for $R$-matrices of exterior powers of the vector representations of $U_q(\hat{\mathfrak{gl}}_N)$ and relate it to the dynamical Weyl group of Tarasov-Varchenko and Etingof-Varchenko, via a Howe ($\mathfrak{gl}_N,\mathfrak{gl}_M$)-duality. As a consequence we obtain a dynamical action of the Weyl group on integrable $U_q\mathfrak{gl}_M$-modules, extending the known action on zero weight spaces.