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

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Abstract

The fusion ring for \(\widehat{\mathfrak {sl}}(n)_m\) 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 \(\widehat{\mathfrak {sl}}(n)_m\) Wess–Zumino–Witten fusion ring associated with the Macdonald polynomials.

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Acknowledgements

Helpful feedback from Stephen Griffeth and constructive remarks made by anonymous referees are gratefully acknowledged. The work of JFvD was supported in part by the Fondo Nacional de Desarrollo Científico y Tecnológico (FONDECYT) Grant # 1210015. TG was supported in part by the NKFIH Grant K134946. This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 795471.}

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  1. Instituto de Matemáticas, Universidad de Talca, Casilla 747, Talca, Chile

    Jan Felipe van Diejen

  2. Bernoulli Institute for Mathematics, Computer Science and Artificial Intelligence, University of Groningen, PO Box 407, 9700 AK, Groningen, The Netherlands

    Tamás Görbe

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  1. Jan Felipe van Diejen
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Diejen, J.F.v., Görbe, T. Elliptic Ruijsenaars difference operators, symmetric polynomials, and Wess–Zumino–Witten fusion rings. Sel. Math. New Ser. 29, 80 (2023). https://doi.org/10.1007/s00029-023-00883-6

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