On the representation type of higher preprojective algebras of type $A$

In this paper, we establish a finite–tame–wild trichotomy for the representation type of $(n+1)$-preprojective algebras of type $A_m$. We also provide a complete classification of such algebras with respect to $\tau$-tilting finiteness. Moreover, for $n\ge 2$, we give a precise characterization of when each indecomposable module is $\tau$-rigid.