Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 21 (2025), 041, 41 pages      arXiv:2410.08657      https://doi.org/10.3842/SIGMA.2025.041

Twisted Fusion Products and Quantum Twisted $Q$-Systems

Mingyan Simon Lin
Singapore Institute of Manufacturing Technology (SIMTech), Agency for Science, Technology and Research (A*STAR), 5 Cleantech Loop, #01-01 CleanTech Two Block B, Singapore 636732, Republic of Singapore

Received October 14, 2024, in final form May 21, 2025; Published online June 10, 2025

Abstract
We obtain a complete characterization of the space of matrix elements dual to the graded multiplicity space arising from fusion products of Kirillov-Reshetikhin modules over special twisted current algebras defined by Kus and Venkatesh, which generalizes the result of Ardonne and Kedem to the special twisted current algebras. We also prove the conjectural identity of $q$-graded fermionic sums by Hatayama et al. for the special twisted current algebras, from which we deduce that the graded tensor product multiplicities of the fusion products of Kirillov-Reshetikhin modules over special twisted current algebras are both given by the $q$-graded fermionic sums, and constant term evaluations of products of solutions of the quantum twisted $Q$-systems obtained by Di Francesco and Kedem.

Key words: twisted $Q$-systems; quantum $Q$-systems; Kirillov-Reshetikhin modules; fusion products.

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