A combinatorial classification of 2-regular simple modules for Nakayama algebras

JOURNAL OF PURE AND APPLIED ALGEBRA

卷:225

Article

Marczinzik, Rene¹  Rubey, Martin²  Stump, Christian³ 

[1] Univ Stuttgart, Inst Algebra & Number Theory, Stuttgart, Germany

[2] TU Wien, Fak Math & Geoinformat, Vienna, Austria

[3] Ruhr Univ Bochum, Fak Math, Bochum, Germany

关键词: Nakayama algebras;    Quiver representation theory;    Homological algebra;    Dyck paths;    Bijective combinatorics;    Combinatorial statistics;   

DOI  :  10.1016/j.jpaa.2020.106520

来源: Elsevier

【 摘 要 】

Enomoto showed for finite dimensional algebras that the classification of exact structures on the category of finitely generated projective modules can be reduced to the classification of 2-regular simple modules. In this article, we give a combinatorial classification of 2-regular simple modules for Nakayama algebras and we use this classification to answer several natural questions such as when there is a unique exact structure on the category of finitely generated projective modules for Nakayama algebras. We also classify 1-regular simple modules, quasi-hereditary Nakayama algebras and Nakayama algebras of global dimension at most two. It turns out that most classes are enumerated by well-known combinatorial sequences, such as Fibonacci, Riordan and Narayana numbers. We first obtain interpretations in terms of the Auslander-Reiten quiver of the algebra using homological algebra, and then apply suitable bijections to relate these to combinatorial statistics on Dyck paths. (C) 2020 Elsevier B.V. All rights reserved.

【 授权许可】

Free