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

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 $(\mathbb{Z}/p^l\mathbb{Z}{)}^n⋊H$ is $\tau$-tilting infinite when $p^l\ge n$ and $\#\mathrm{IBr}H\ge \min \{p,3\}$, 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 $p^l\ge n$ and $H$ is a $p^{\prime }$-subgroup of ${\mathfrak{S}}_n$, $\tau$-tilting finiteness of a group algebra of a group $(\mathbb{Z}/p^l\mathbb{Z}{)}^n⋊H$ is determined by its $p$-hyperfocal subgroup.

15 pages. arXiv admin note: text overlap with arXiv:2405.10021