Twisted Fusion Products and Quantum Twisted $Q$-Systems

TL;DR

This paper characterizes the dual space of matrix elements related to fusion products of Kirillov-Reshetikhin modules over special twisted current algebras, proving conjectural identities and connecting graded multiplicities to quantum $Q$-systems.

Contribution

It provides a complete description of the dual space for these fusion products and proves the conjectural fermionic sum identities for the first time.

Findings

Graded tensor product multiplicities are given by $q$-graded fermionic sums.

Proves the conjectural identities of Hatayama et al. for special twisted current algebras.

Connects multiplicities to constant term evaluations of quantum twisted $Q$-system solutions.

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.