$\tau$-Tilting finiteness of group algebras over generalized symmetric groups

TL;DR

This paper investigates the conditions under which group algebras over generalized symmetric groups are $ au$-tilting finite or infinite, linking algebraic properties to Cartan matrices and subgroup structures.

Contribution

It establishes criteria for $ au$-tilting finiteness and infiniteness of weakly symmetric and selfinjective algebras, especially for group algebras of certain semidirect products involving symmetric groups.

Findings

Weakly symmetric $ au$-tilting finite algebras have positive definite Cartan matrices.

Group algebra of $(Z/p^lZ)^n times H$ is $ au$-tilting infinite if $p^l geq n$ or $ ext{IBr} hinspace H$ is large.

Finiteness depends on the $p$-hyperfocal subgroup when $H$ is a $p'$-subgroup of $rak{S}_n$.

Abstract

In this paper, we show that weakly symmetric $\tau$-tilting finite algebras have positive definite Cartan matrices, which implies that we can prove $\tau$-tilting infiniteness of weakly symmetric algebras by calculating their Cartan matrices. Similarly, we obtain the condition on Cartan matrices that selfinjective algebras are $\tau$-tilting infinite. By applying this result, we show that a group algebra of $(\mathbb{Z}/p^l\mathbb{Z})^n\rtimes H$ is $\tau$-tilting infinite when $p^l\geq n$ and $\#\mathrm{IBr}\,H\geq\min\{p,3\}$, where $p>0$ is the characteristic of the ground field, $H$ is a subgroup of the symmetric group $\mathfrak{S}_n$ of degree $n$, the action of $H$ permutes the entries of $(\mathbb{Z}/p^l\mathbb{Z})^n$, and $\mathrm{IBr}\,H$ denotes the set of irreducible $p$-Brauer characters of $H$. Moreover, we show that under the assumption that $p^l\geq n$ and $H$ is a $p'$-subgroup of $\mathfrak{S}_n$, $\tau$-tilting finiteness of a group algebra of a group $(\mathbb{Z}/p^l\mathbb{Z})^n\rtimes H$ is determined by its $p$-hyperfocal subgroup.