Classification of Auslander-Gorenstein monomial algebras: The acyclic case

TL;DR

This paper classifies Auslander-Gorenstein acyclic monomial algebras using Bruhat factorisation of Coxeter matrices, revealing new structural insights and answering longstanding questions in the field.

Contribution

It provides a linear algebraic classification of these algebras via Bruhat factorisation, linking homological properties to matrix decompositions and resolving open questions.

Findings

A monomial acyclic quiver algebra is Auslander regular iff its Coxeter matrix has a Bruhat factorisation with $U_1$ as the identity.

Linear Nakayama algebras are always Auslander regular under these conditions.

General Auslander regular acyclic quiver algebras are echelon-independent, confirming a recent conjecture.

Abstract

We give a linear algebraic classification of Auslander regular acyclic monomial algebras via the Bruhat factorisation of the Coxeter matrix. Namely, we show under mild assumptions that a monomial acyclic quiver algebra is Auslander regular if and only if its Coxeter matrix $C$ has a Bruhat factorisation $U_1 P U_2$ with $U_1$ the identity matrix. In particular, this holds without restrictions for linear Nakayama algebras and we use the Bruhat decomposition to answer a question raised by Ringel by showing that his homological permutation coincides with the permutation coming from the Bruhat factorisation of the Coxeter matrix. We also use our methods to show that general Auslander regular acyclic quiver algebras are echelon-independent, proving a conjecture of Defant-Jiang-Marczinzik-Segovia-Speyer-Thomas-Williams, and we answer another question by Ringel on the delooping level of simple modules over Nakayama algebras.