$\tau$-tilting finite gentle algebras are representation-finite

Abstract

We show that a gentle algebra over a field is $\tau$-tilting finite if and only if it is representation-finite. The proof relies on the "brick-$\tau$-tilting correspondence" of Demonet-Iyama-Jasso and on a combinatorial analysis.

7 pages. v2: updated references. v3: Fixed proof of Proposition 3.3