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Howe Duality and Dynamical Weyl Group.

Rea Dalipi¹, Giovanni Felder², Anfisa Gurenkova³

¹Department of Mathematics, University of Geneva, Rue du Conseil-Général 7-9, 1205 Geneva, Switzerland.

Transformation Groups

|May 25, 2026

Summary

This study presents a fermionic formula for R-matrices of exterior powers of quantum group representations. It connects to dynamical Weyl groups and yields R-matrices for Fock spaces, extending Weyl group actions on integrable modules.

Keywords:

Dynamical Weyl group Howe duality Quantum groups

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Area of Science:

Quantum Algebra
Representation Theory
Mathematical Physics

Background:

The study builds upon existing work in quantum group theory and representation theory.
It addresses the need for explicit formulas for R-matrices in specific representations.

Purpose of the Study:

To derive a fermionic formula for R-matrices of exterior powers of vector representations of U_q(gl_N).
To establish a connection between these R-matrices and the dynamical Weyl group.
To investigate the implications for Fock spaces and integrable modules.

Main Methods:

Utilizing a fermionic formula approach.
Applying Howe duality for (gl_N, gl_M).
Analyzing the limit as N approaches infinity.

Main Results:

A fermionic formula for R-matrices of exterior powers of U_q(gl_N) vector representations is obtained.
The formula is related to the dynamical Weyl group of Tarasov-Varchenko and Etingof-Varchenko.
R-matrices for Fock spaces are derived in the limit N -> infinity.
A dynamical action of the Weyl group on integrable U_q(gl_M)-modules is established.

Conclusions:

The derived R-matrices provide new insights into the structure of quantum groups and their representations.
The results extend the known action of the Weyl group on integrable modules.
The findings have potential implications for various areas of mathematical physics, including integrable systems and quantum field theory.