Howe duality and dynamical Weyl group

2022/08/27 by Dalipi, Rea, Felder, Giovanni
#17B37 #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2208.13055

Abstract

We give a fermionic formula for R-matrices of exterior powers of the vector representations of U_q(\widehat \mathfrakgl_N) and relate it to the dynamical Weyl group of Tarasov--Varchenko and Etingof--Varchenko, via a Howe (\mathfrakgl_N,\mathfrakgl_M)-duality. In the limit N→∞ 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\mathfrakgl_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 \mathfrakgl_M by a general symmetrizable Kac--Moody algebra.

Related

• Orthogonal Howe duality and dynamical (split) symmetric pairs
• Duality on Fock spaces and combinatorial energy functions
• Dynamical Weyl groups and applications
• The Howe duality and the Projective Representations of Symmetric Groups
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