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arXiv · 1811.05846

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

Rene Marczinzik·Martin Rubey·Christian Stump
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Abstract

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.

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Rene Marczinzik, Martin Rubey, Christian Stump. 2018-11-14. A combinatorial classification of 2-regular simple modules for Nakayama algebras. https://arxiv.org/abs/1811.05846

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