Title: Graded fusion product multiplicities and quantum cluster algebras


Q-systems are rational recursion relations satisfied by special characters of simple Lie algebras. They can be viewed as mutations in a cluster algebra. They are closely connected to fermionic formulas for the multiplicities of the irreducible components in the tensor product of Kirillov-Reshetikhin modules. The graded version of these formulas, which appeared in the 80's in the context of the Bethe ansatz, has an interpretation as a graded tensor product multiplicities given by Feigin and Loktev in 2000.
The system which is naturally associated to these formulas is the quantum Q-system, which is a mutation in a quantum cluster algebra. This system was solved in recent work [joint with Di Francesco]. The graded fermionic formulas are a natural application of this system.