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

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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.