On the global dimension of Nakayama algebras
Combinatorics
2025-03-25 v1 Representation Theory
Abstract
We study the global dimension of Nakayama algebras. In the case of linear Nakayama algebras, which are in canonical bijection to Dyck paths, we show that the global dimension has the same distribution as the height of Dyck paths. For cyclic Nakayama algebras an explicit classification of finite global dimension is not known. However, we show that in certain special cases cyclic Nakayama algebras with finite global dimension can again be interpreted as Dyck paths. In particular, we show that there is a natural bijection between sincere Nakayama algebras and Dyck paths. In this case, we find that the global dimension is in fact twice the bounce count of the corresponding Dyck path.
Cite
@article{arxiv.2503.18415,
title = {On the global dimension of Nakayama algebras},
author = {Viktória Klász and René Marczinzik and Anton Mellit and Martin Rubey and Christian Stump},
journal= {arXiv preprint arXiv:2503.18415},
year = {2025}
}
Comments
22 pages