Kirillov-Reshetikhin conjecture : the general case

Abstract

We prove the Kirillov-Reshetikhin (KR) conjecture in the general case : for all twisted quantum affine algebras we prove that the characters of KR modules solve the twisted Q-system and we get explicit formulas for the character of their tensor products (the untwisted simply-laced case was treated by Najakima, and the untwisted case by the author). The proof is uniform and provides several new developments for the representation theory of twisted quantum affine algebras, including twisted Frenkel-Reshetikhin q-characters (expected by Frenkel-Reshetikhin and Frenkel-Mukhin). We also prove the twisted T-system. As an application we get explicit formulas for the twisted q-characters of fundamental representations for all types, including the formulas for types D_4^{(3)}, E_6^{(2)} conjectured by Reshetikhin. We prove the formulas for KR modules in types A_n^{(2)} and D_4^{(3)} conjectured by Kuniba-Suzuki. Eventually our results imply the conjectural branching rules [HKOTT] to the quantum subalgebra of finite type.