The ring of perfect p-permutation bimodules for blocks with cyclic defect groups

2024/08/08 by Boltje, Robert, Monteiro, Nariel
#19A22 #20C20 #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2408.04134

Abstract

Let B be a block algebra of a group algebra FG of a finite group G over a field F of characteristic p>0. This paper studies ring theoretic properties of the representation ring T^Δ(B,B) of perfect p-permutation (B,B)-bimodules and properties of the k-algebra k⊗_ℤ T^Δ(B,B), for a field k. We show that if the Cartan matrix of B has 1 as an elementary divisor then [B] is not primitive in T^Δ(B,B). If B has cyclic defect groups we determine a primitive decomposition of [B] in T^Δ(B,B). Moreover, if k is a field of characteristic different from p and B has cyclic defect groups of order pⁿ we describe k⊗_ℤ T^Δ(B,B) explicitly as a direct product of a matrix algebra and n group algebras.

Related

• p-permutation equivalences between blocks of group algebras
• On the source algebra equivalence class of blocks with cyclic defect groups, I
• On the source algebra equivalence class of blocks with cyclic defect groups, II
• The primitive idempotents of the p-permutation ring
• The source permutation module of a block of a finite group algebra