A pentagon of identities, graded tensor products and the Kirillov-Reshetikhin conjecture

Abstract: This paper provides a brief review of the relations between the Feigin-Loktev conjecture on the dimension of graded tensor products of $\g[t]$-modules, the Kirillov-Reshetikhin conjecture, the combinatorial ``M=N" conjecture, their proofs for all simple Lie algebras, and a pentagon of identities which results from the proof.