Classifying tilting modules over the Auslander algebras of radical square zero Nakayama algebras

TL;DR

This paper classifies tilting modules over Auslander algebras derived from radical square zero Nakayama algebras, revealing their structure and enumeration based on algebra properties.

Contribution

It provides a complete classification of indecomposable summands of tilting modules and counts the total tilting modules depending on whether the algebra is self-injective.

Findings

Indecomposable summands are either simple or projective

Number of tilting modules is 2^n if self-injective

Number of tilting modules is 2^{n-1} otherwise

Abstract

Let $\Lambda$ be a radical square zero Nakayama algebra with $n$ simple modules and let $\Gamma$ be the Auslander algebra of $\Lambda$. Then every indecomposable direct summand of a tilting $\Gamma$-module is either simple or projective. Moreover, if $\Lambda$ is self-injective, then the number of tilting $\Gamma$-modules is $2^n$; otherwise, the number of tilting $\Gamma$-modules is $2^{n-1}$.