Bricks over preprojective algebras and join-irreducible elements in Coxeter groups

JOURNAL OF PURE AND APPLIED ALGEBRA

卷:226

Article

Asai, Sota¹ 

[1] Osaka Univ, Grad Sch Informat Sci & Technol, Dept Pure & Appl Math, Suita, Osaka 5650871, Japan

关键词: Bricks;    tau-Tilting theory;    Preprojective algebras;    Coxeter groups;    Lattices;    Canonical join representations;   

DOI  :  10.1016/j.jpaa.2021.106812

来源: Elsevier

【 摘 要 】

A (semi)brick over an algebra A is a module S such that its endomorphism ring End(A)(S) is a (product of) division algebra. For each Dynkin diagram Delta, there is a bijection from the Coxeter group W of type Delta to the set of semibricks over the preprojective algebra Pi of type Delta, which is restricted to a bijection from the set of join-irreducible elements of W to the set of bricks over Pi. This paper is devoted to giving an explicit description of these bijections in the case Delta = A(n) or D-n. First, for each join-irreducible element w is an element of W, we describe the corresponding brick S(w) in terms of Young diagram-like notation. Next, we determine the canonical join representation w = V-i=1(m) w(i) of an arbitrary element w is an element of W based on Reading's work, and prove that circle plus(m)(i=1) S(w(i)) is the semibrick corresponding to w. (C) 2021 Elsevier B.V. All rights reserved.

【 授权许可】

Free