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

TL;DR

This paper investigates the $ au$-tilting finiteness of certain finite-dimensional algebras, including symmetric algebras of polynomial growth, revealing invariance under derived equivalence and connections with representation-finiteness.

Contribution

It establishes $ au$-tilting finiteness for symmetric algebras of polynomial growth and shows its preservation under derived equivalence, also relating it to representation-finiteness in specific algebra classes.

Findings

Symmetric algebras of polynomial growth are $ au$-tilting finite.

Derived equivalence preserves $ au$-tilting finiteness in these algebras.

For $0$-Hecke and $0$-Schur algebras, $ au$-tilting finiteness coincides with representation-finiteness.

Abstract

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.