 Brauer tree algebra

The basic algebra of a split block with cyclic defect is a Brauer tree algebra: its simple modules are indexed by the edges of a tree, with a distinguished vertex carrying the exceptional multiplicity. For a one-edge tree of multiplicity $m$, its basic algebra is $k[t]/(t^{m+1})$. Its ideals form the chain of powers of $t$, so its unique indecomposable projective is uniserial of length $m+1$.