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

David Hernandez, Hironori Oya

研究成果: Article

1 引用 (Scopus)

抄録

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.

元の言語English
ページ(範囲)192-272
ページ数81
ジャーナルAdvances in Mathematics
347
DOI
出版物ステータスPublished - 2019 4 30

ASJC Scopus subject areas

  • Mathematics(all)

これを引用

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title = "Quantum Grothendieck ring isomorphisms, cluster algebras and Kazhdan-Lusztig algorithm",
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.",
keywords = "Dual canonical bases, Kazhdan-Lusztig algorithm, Quantum affine algebras, Quantum cluster algebras, Quantum Grothendieck rings, T-systems",
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T1 - Quantum Grothendieck ring isomorphisms, cluster algebras and Kazhdan-Lusztig algorithm

AU - Hernandez, David

AU - Oya, Hironori

PY - 2019/4/30

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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

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