Howe Duality and Dynamical Weyl Group

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