Classifying torsion classes for algebras with radical square zero via sign decomposition

TL;DR

This paper introduces a sign-decomposition method to classify torsion classes of algebras with radical square zero, linking them to hereditary algebras and applying this to count support τ-tilting modules in specific algebra classes.

Contribution

It develops a novel sign-decomposition approach for torsion classes and establishes a bijection with hereditary algebras, enhancing understanding of their structure and enumeration.

Findings

Bijection between torsion classes of algebra with radical square zero and hereditary algebras.

Preservation of functorially finite property under the bijection.

Exact counts of support τ-tilting modules for Brauer line and odd-cycle algebras.

Abstract

To study the set of torsion classes of a finite dimensional basic algebra, we use a decomposition, called sign-decomposition, parametrized by elements of $\{\pm1\}^n$ where $n$ is the number of simple modules. If $A$ is an algebra with radical square zero, then for each $\epsilon \in \{\pm1\}^n$ there is a hereditary algebra $A_{\epsilon}^!$ with radical square zero and a bijection between the set of torsion classes of $A$ associated to $\epsilon$ and the set of faithful torsion classes of $A_{\epsilon}^!$. Furthermore, this bijection preserves the property of being functorially finite. As an application in $\tau$-tilting theory, we prove that the number of support $\tau$-tilting modules over Brauer line algebras (resp. Brauer odd-cycle algebras) having $n$ edges is $\binom{2n}{n}$ (resp. $2^{2n-1}$).