The Kirillov-Reshetikhin conjecture and solutions of T-systems

Abstract

We prove the Kirillov-Reshetikhin conjecture for all untwisted quantum affine algebras: we prove that the characters of Kirillov-Reshetikhin modules solve the Q-system and we give an explicit formula for the character of their tensor products. In the proof we show that Kirillov-Reshetikhin modules are special in the sense of monomials and that their q-characters solve the T-system (functional relations appearing in the study of solvable lattice models). Moreover we prove that the T-system can be written in the form of an exact sequence. For simply-laced cases, these results were proved by Nakajima in [H. Nakajima, Quiver Varieties and t-Analogs of q-Characters of Quantum Affine Algebras, Ann. Math. 160 (2004), 1057–1097.], [H. Nakajima, t-analogs of q-characters of Kirillov-Reshetikhin modules of quantum affine algebras, Represent. Th. 7 (2003), 259–274.] with geometric arguments (main result of [H. Nakajima, Quiver Varieties and t-Analogs of q-Characters of Quantum Affine Algebras, Ann. Math. 160 (2004), 1057–1097.]) which are not available in general. The proof we use is different and purely algebraic, and so can be extended uniformly to non simply-laced cases.