Positivity and periodicity of $Q$-systems in the WZW fusion ring

TL;DR

This paper investigates the positivity and periodicity of solutions to $Q$-systems in the WZW fusion ring, providing proofs in specific cases and constructing positive solutions for classical types, thereby confirming related conjectures.

Contribution

It introduces new conjectures on positivity and periodicity of $Q$-systems and proves some of these conjectures, advancing understanding of fusion rings and $Q$-system solutions.

Findings

Proof of positivity and periodicity in some cases

Construction of positive solutions for classical types

Verification of conjectures by Kirillov and colleagues

Abstract

We study properties of solutions of $Q$-systems in the WZW fusion ring obtained by the Kirillov-Reshetikhin modules. We make a conjecture about their positivity and periodicity and give a proof of it in some cases. We also construct a positive solution of the level $k$ restricted $Q$-system of classical types in the fusion rings. As an application, we prove some conjectures of Kirillov and Kuniba-Nakanishi-Suzuki on the level $k$ restricted $Q$-systems.