$\tau$-tilting Finiteness of Group Algebras over Generalized Symmetric Groups

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 sufficient condition on Cartan matrices that selfinjective algebras are $\tau$-tilting infinite. By applying this result, we show that a group algebra of a semidirect product $(\mathbb{Z}/p^{l} \mathbb{Z})^{n} \rtimes H$ is $\tau$-tilting infinite when $p^{l} \geq n$ and $\# \operatorname{IBr}H \geq \min \{p,3\}$, 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 $\operatorname{IBr}H$ 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 $(\mathbb{Z}/p^{l} \mathbb{Z})^{n} \rtimes H$ is determined by its $p$-hyperfocal subgroup.