Elliptic Ruijsenaars difference operators, symmetric polynomials, and Wess-Zumino-Witten fusion rings

TL;DR

This paper introduces a deformation of the Wess-Zumino-Witten fusion ring using elliptic Ruijsenaars difference operators, connecting symmetric polynomials, eigenpolynomials, and fusion rules with new algebraic structures.

Contribution

It presents a novel deformation of the fusion ring linked to elliptic Ruijsenaars operators, extending the algebraic framework of WZW models and symmetric polynomials.

Findings

Deformation of the fusion ring via elliptic Ruijsenaars operators

Pieri rule governs Littlewood-Richardson coefficients in this setting

Recovery of the refined fusion ring in the trigonometric limit

Abstract

The fusion ring for $\widehat{\mathfrak{su}}(n)_m$ Wess-Zumino-Witten conformal field theories is known to be isomorphic to a factor ring of the ring of symmetric polynomials presented by Schur polynomials. We introduce a deformation of this factor ring associated with eigenpolynomials for the elliptic Ruijsenaars difference operators. The corresponding Littlewood-Richardson coefficients are governed by a Pieri rule stemming from the eigenvalue equation. The orthogonality of the eigenbasis gives rise to an analog of the Verlinde formula. In the trigonometric limit, our construction recovers the refined $\widehat{\mathfrak{su}}(n)_m$ Wess-Zumino-Witten fusion ring associated with the Macdonald polynomials.