Howe Duality and Dynamical Weyl Group

Transformation Groups. 2024.

Felder G., Dalipi R.

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.

Keywords: R-matrix, quantum groups, Howe duality, dynamical Weyl groups