p-permutation equivalences between blocks of group algebras

2020/07/17 by Robert Boltje, Boltje, Robert, Philipp Perepelitsky +1 · 5 citations
Mathematics · Computer Science · #Finite Group Theory Research #Algebraic structures and combinatorial models #Coding theory and cryptography

paper · pdf · doi:10.48550/arxiv.2007.09253

Abstract

We extend the notion of a p-permutation equivalence between two p-blocks A and B of finite groups G and H, from the definition in [Boltje-Xu 2008] to a virtual p-permutation bimodule whose components have twisted diagonal vertices. It is shown that various invariants of A and B are preserved, including defect groups, fusion systems, and Külshammer-Puig classes. Moreover it is shown that p-permutation equivalences have additional surprising properties. They have only one constituent with maximal vertex and the set of p-permutation equivalences between A and B is finite (possibly empty). The paper uses new methods: a consequent use of module structures on subgroups of G× H arising from Brauer constructions which in general are not direct product subgroups, the necessary adaptation of the notion of tensor products between bimodules, and a general formula (stated in these new terms) for the Brauer construction of a tensor product of p-permutation bimodules.

Cited by

• Fixed points of extended tensor products
• Endotrivial complexes
• On RoCK blocks of double covers of symmetric and alternating groups and the refined Broué conjecture
• The Euler characteristic of an endotrivial complex
• On Alperin's conjecture and functorial equivalence of blocks

Related

• Fixed points of extended tensor products
• The ring of perfect p-permutation bimodules for blocks with cyclic defect groups
• Splendid and perverse equivalences
• On the source algebra equivalence class of blocks with cyclic defect groups, II
• On p-permutation equivalences between direct products of blocks