Let $A$ be a finite dimensional algebra over an algebraically closed field. I'm trying to better understand the difference between $A$ being representation infinite and $A$ being $\tau$-tilting infinite when $A$ is a string algebra. So, if $A$ is representation finite, then $A$ is also $\tau$-tilting finite.


I'm wondering what is an example of a string algebra $A$ that is representation infinite but $\tau$-tilting finite.



Rep infinite, but $\tau$-tilting finite
I think the group algebra of a dihedral $2$-group in characteristic two, mod its socle, is an example. The smallest of these is $k\langle x,y\rangle/(x^2,y^2,xy,yx)$. See Plamondon, "$\tau$-Tilting finite gentle algebras are representation finite."


