Fusion products of Kirillov–Reshetikhin modules and fermionic multiplicity formulas

We give a complete description of the graded multiplicity space which appears in the Feigin–Loktev fusion product of graded Kirillov–Reshetikhin modules for all simple Lie algebras. This construction is used to obtain an upper bound formula for the fusion coefficients in these cases. The formula generalizes the case of of our previous paper, where the multiplicities are generalized Kostka polynomials. In the case of other Lie algebras, the formula is the fermionic side of the X=M conjecture. In the cases where the Kirillov–Reshetikhin conjecture, regarding the decomposition formula for tensor products of KR-modules, has been proven in its original, restricted form, our result provides a proof of the conjectures of Feigin and Loktev regarding the fusion product multiplicities.