A Cluster tilting module for a representation-infinite block of a group algebra

Abstract

Let G = SL(2, 5) be the special linear group of 2 × 2-matrices with coefficients in the field with 5 elements. We show that the principal block over a splitting field K of characteristic two of the group algebra KG has a 3-cluster tilting module. This gives the first example of a representation-infinite block of a group algebra having a cluster tilting module and answers a question by Erdmann and Holm.

Some cluster tilting modules for weighted surface algebras

“…First, by dimension shift Ext 1 (U ν , U bj ) ∼ = Ext 1 (S ν , S bj ) = 0 for any ν of valency 1, from the quiver. Next, consider Ext 2 (U ν , X), by applying the functor Hom(−, X) to the second exact sequence in (2). We have Hom(S µ , U bj ) = 0 (the socle of U bi is always some S a ), and hence Ext 2 (U ν , X) = 0.…”

“…From the first sequence we get 0 → Hom(U di , X) → Xe ai+1 → 0 → Ext 1 (U di , X) → 0 and the ext space is zero. Now consider (−, X) applied to sequences in (2). The second sequence gives Hom(Ω −1 (S bi ), X) ∼ = Xe bi and Ext 1 (Ω −1 (S bi ), X) = 0.…”

“…When the characteristic of K is 2 these occur as the basic algebras of blocks of finite groups. Very recently B. Böhmler and R. Marcinczik proved using computer calculations that for k = 2, it has a 3-cluster tilting module (see [2]). Much of this note was written five years ago, when talking to Idun Reiten about [3], and it was extended first when spherical algebras had been discovered, and then again, inspired by an email from R. Marcinczik (for which I am grateful).…”