Tame blocks

A finite dimensional [math]k[/math]-algebra [math]A[/math] is said to have finite representation type if there are only finitely many isomorphism classes of indecomposable modules. Algebras of infinite representation type are split into two cases: tame and wild. For definitions see Section I.4 of [Er90], but tame essentially means that almost all modules of a given dimension fit into finitely many one-parameter families and wild means that the module category is comparable to that for [math]k\langle X,Y \rangle[/math]. The properties of having finite or infinite representation type and of being tame or wild are all Morita invariants.

Let [math]B[/math] be a block of [math]kG[/math] for a finite group [math]G[/math] with defect group [math]D[/math]. Then

• [math]B[/math] has finite representation type if and only if [math]D[/math] is cyclic.
• [math]B[/math] is tame if and only if [math]D[/math] contains no noncyclic abelian [math]p[/math]-subgroup of order greater than four. Equivalently, [math]D[/math] is generalized quaternion, dihedral or semidihedral.
• Otherwise, [math]B[/math] is wild.

In a series of papers and her book [Er90], Erdmann describes the basic algebra of tame type (see page vi of [Er90] for a definition), and in most cases describes which of these occur as basic algebras for blocks of finite groups.