  We give a fermionic formula for $R$-matrices of exterior powers of the vector
representations of $U_q(\widehat{ \mathfrak{gl}}_N)$ and relate it to the
dynamical Weyl group of Tarasov--Varchenko and Etingof--Varchenko, via a Howe
($\mathfrak{gl}_N,\mathfrak{gl}_M)$-duality. 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 $U_q\mathfrak{gl}_M$-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
$\mathfrak{gl}_M$ by a general symmetrizable Kac--Moody algebra.
