On $\tau$-tilting finiteness of symmetric algebras of polynomial growth

In this paper, we report on the $\tau$-tilting finiteness of some classes of finite-dimensional algebras over an algebraically closed field, including symmetric algebras of polynomial growth, $0$-Hecke algebras and $0$-Schur algebras. Consequently, we find that derived equivalence preserves the $\tau$-tilting finiteness over symmetric algebras of polynomial growth, and self-injective cellular algebras of polynomial growth are $\tau$-tilting finite. Furthermore, the representation-finiteness and $\tau$-tilting finiteness over $0$-Hecke algebras and $0$-Schur algebras (with few exceptions) coincide.