τ-Tilting finiteness of group algebras over generalized symmetric groups

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