Quasi-Hereditary Orderings of Nakayama Algebras

TL;DR

This paper provides a simple criterion for identifying quasi-hereditary orderings in Nakayama algebras, classifies all such orderings, and proves the $q$-ordering conjecture for this class.

Contribution

It introduces a criterion for $q$-orderings in Nakayama algebras, enabling full classification and explicit calculation of $q(A)$, and confirms the conjecture for these algebras.

Findings

Established a criterion for $q$-orderings in Nakayama algebras.

Classified all $q$-orderings for Nakayama algebras.

Proved the $q$-ordering conjecture for Nakayama algebras.

Abstract

Let $A$ be an algebra with iso-class of simple modules $\mathcal{S}$ of cardinality $n$. A total ordering on $\mathcal{S}$ making every Weyl module Schurian and every indecomposable projective module filtered by the Weyl modules is called to be a quasi-hereditary ordering or $q$-ordering on $A$ and $A$ is a quasi-hereditary algebra under this ordering. The number of $q$-orderings on $A$ is denoted by $q(A)$. To determine whether an ordering on $\mathcal{S}$ is a $q$-ordering is a hard problem. A famous result due to Dlab and Ringel is that $A$ is hereditary if and only if every ordering is a $q$-ordering, equivalently, $q(A)=n!$. The twenty-years old $q$-ordering conjecture claims that $q(A)\le\dfrac{2}{3}n!$. The present paper proves a very simple criterion for $q$-orderings when $A$ is a Nakayama algebra. This criterion is applied to getting a full classification of all $q$-orderings of $A$ and an explicit iteration formula for $q(A)$, and also a positive proof of the $q$-ordering conjecture for Nakayama algebras.