We establish ring isomorphisms between quantum Grothendieck rings of certain remarkable monoidal categories C Q,B n and C Q,A 2n−1 of finite-dimensional representations of quantum affine algebras of types B n (1) and A 2n−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 C Q,B n are given by the specialization of certain analogues of Kazhdan-Lusztig polynomials and the coefficients of these polynomials are positive.
ASJC Scopus subject areas
- 数学 (全般)