Quantum Grothendieck ring isomorphisms, cluster algebras and Kazhdan-Lusztig algorithm

Abstract

We establish ring isomorphisms between quantum Grothendieck rings of certain remarkable monoidal categories CQ,Bn and CQ,A2n−1 of finite-dimensional representations of quantum affine algebras of types Bn(1) and A2n−1(1), respectively. Our proof relies in part on the corresponding quantum cluster algebra structures. Moreover, we prove that our isomorphisms specialize at t=1 to the isomorphisms of (classical) Grothendieck rings obtained recently by Kashiwara, Kim and Oh by other methods. As a consequence, we prove a conjecture formulated by the first author in 2002: the multiplicities of simple modules in standard modules in CQ,Bn are given by the specialization of certain analogues of Kazhdan-Lusztig polynomials and the coefficients of these polynomials are positive.