Cluster algebras of finite type via Coxeter elements and Demazure crystals of type A

JOURNAL OF ALGEBRA

卷:538

Article

Kanakubo, Yuki¹  Nakashima, Toshiki¹ 

[1] Sophia Univ, Div Math, Chiyoda Ku, Kioicho 7-1, Tokyo 1028554, Japan

关键词: Cluster algebras;    Double Bruhat cells;    Additive categorifications;    Preprojective algebras;    Crystals;    Monomial realizations;   

DOI  :  10.1016/j.jalgebra.2019.08.001

来源: Elsevier

【 摘 要 】

Let C be a simply connected simple algebraic group over C, B and B- be its two opposite Borel subgroups. For two elements u, v of the Weyl group W, it is known that the coordinate ring C[G(u,v)] of the double Bruhat cell G(u,v) = BuB boolean AND B- vB(-) is isomorphic to a cluster algebra A(i)(C) [2, 12]. In the case u = e, v = c(2) (c is a Coxeter element), the algebra C[G(e,c2)] has only finitely many cluster variables. In this article, for G = SLr+1(C), we obtain explicit forms of all the cluster variables in C[G(e,c2)] by considering its additive categorification via preprojective algebras, and describe them in terms of monomial realizations of Demazure crystals. (C) 2019 Elsevier Inc. All rights reserved.

【 授权许可】

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