2018/11/14 by René Marczinzik, Marczinzik, Rene, Martin Rubey +3 · 1 citation
Mathematics · #Algebraic structures and combinatorial models #Advanced Combinatorial Mathematics #Commutative Algebra and Its Applications
paper · pdf · doi:10.48550/arxiv.1811.05846
Enomoto showed for finite dimensional algebras that the classification of\nexact structures on the category of finitely generated projective modules can\nbe reduced to the classification of 2-regular simple modules. In this article,\nwe give a combinatorial classification of 2-regular simple modules for Nakayama\nalgebras and we use this classification to answer several natural questions\nsuch as when there is a unique exact structure on the category of finitely\ngenerated projective modules for Nakayama algebras. We also classify 1-regular\nsimple modules, quasi-hereditary Nakayama algebras and Nakayama algebras of\nglobal dimension at most two. It turns out that most classes are enumerated by\nwell-known combinatorial sequences, such as Fibonacci, Riordan and Narayana\nnumbers. We first obtain interpretations in terms of the Auslander-Reiten\nquiver of the algebra using homological algebra, and then apply suitable\nbijections to relate these to combinatorial statistics on Dyck paths.\n