3 problems

Connected AR-quiver conjecture. If the Auslander–Reiten quiver of $A$ is connected, then $A$ is representation-finite.

Auslander–Reiten–Smalø conjecture. If $A$ is of infinite representation type, then $\Gamma_A$ has infinitely many connected components.

Let $w_0$ be the longest element of the relevant Weyl group, let $[\widetilde{w}_0]$ denote a commutation equivalence class of reduced expressions of $w_0$, and let an $r$-cluster…