### Abstract

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.

Original language | English |
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Pages (from-to) | 192-272 |

Number of pages | 81 |

Journal | Advances in Mathematics |

Volume | 347 |

DOIs | |

Publication status | Published - 2019 Apr 30 |

### Keywords

- Dual canonical bases
- Kazhdan-Lusztig algorithm
- Quantum Grothendieck rings
- Quantum affine algebras
- Quantum cluster algebras
- T-systems

### ASJC Scopus subject areas

- Mathematics(all)