### 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 |
---|---|

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 affine algebras
- Quantum cluster algebras
- Quantum Grothendieck rings
- T-systems

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**Quantum Grothendieck ring isomorphisms, cluster algebras and Kazhdan-Lusztig algorithm.** / Hernandez, David; Oya, Hironori.

Research output: Contribution to journal › Article

*Advances in Mathematics*, vol. 347, pp. 192-272. https://doi.org/10.1016/j.aim.2019.02.024

}

TY - JOUR

T1 - Quantum Grothendieck ring isomorphisms, cluster algebras and Kazhdan-Lusztig algorithm

AU - Hernandez, David

AU - Oya, Hironori

PY - 2019/4/30

Y1 - 2019/4/30

N2 - 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.

AB - 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.

KW - Dual canonical bases

KW - Kazhdan-Lusztig algorithm

KW - Quantum affine algebras

KW - Quantum cluster algebras

KW - Quantum Grothendieck rings

KW - T-systems

UR - http://www.scopus.com/inward/record.url?scp=85062144884&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85062144884&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2019.02.024

DO - 10.1016/j.aim.2019.02.024

M3 - Article

AN - SCOPUS:85062144884

VL - 347

SP - 192

EP - 272

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

ER -