Homological algebra of Nakayama algebras and 321avoiding permutations
Abstract
Linear Nakayama algebras over a field $K$ are in natural bijection to Dyck paths and Dyck paths are in natural bijection to 321avoiding bijections via the BilleyJockuschStanley bijection. Thus to every 321avoiding permutation $\pi$ we can associate in a natural way a linear Nakayama algebra $A_{\pi}$. We give a homological interpretation of the fixed points statistic of 321avoiding permutations using Nakayama algebras with a linear quiver. We furthermore show that the space of selfextension for the Jacobson radical of a linear Nakayama algebra $A_{\pi}$ is isomorphic to $K^{\mathfrak{s}(\pi)}$, where $\mathfrak{s}(\pi)$ is defined as the cardinality $k$ such that $\pi$ is the minimal product of transpositions of the form $s_i=(i,i+1)$ and $k$ is the number of distinct $s_i$ that appear.
 Publication:

arXiv eprints
 Pub Date:
 April 2022
 DOI:
 10.48550/arXiv.2204.13764
 arXiv:
 arXiv:2204.13764
 Bibcode:
 2022arXiv220413764C
 Keywords:

 Mathematics  Combinatorics;
 Mathematics  Representation Theory;
 16G10;
 18G20
 EPrint:
 15 pages, 2 figures, 8 diagrams