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

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.