Howe duality and dynamical Weyl group

TL;DR

This paper introduces a fermionic formula for R-matrices related to exterior powers of vector representations of quantum affine algebras, connecting it to the dynamical Weyl group via Howe duality, and extends Weyl group actions to broader modules.

Contribution

It provides a new fermionic formula for R-matrices and links it to the dynamical Weyl group through Howe duality, also extending Weyl group actions to integrable modules.

Findings

Fermionic formula for R-matrices of exterior powers

Connection between R-matrices and dynamical Weyl group via Howe duality

Extension of Weyl group action to integrable modules and Kac--Moody algebras

Abstract

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.