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.
Categories: math.RT math.GR math.RA