Local fusions in block source algebras
type: Item
instance of: scholarly article
title: Local fusions in block source algebras
zbMATH Open document ID: 0606.20016
DOI: 10.1016/0021-8693(86)90221-8
author: Lluis Puig
published in: Journal of Algebra
publication date: 1986-01-01T00:00:00Z
review text: Almost three decades ago J. A. Green introduced vertices and sources to study the structure of finite groups G and their group rings. Guided by Green's profound investigations the author considers the general concept of an interior G-algebra A to define so called source algebras of A. He writes in his introduction: Obviously the most interesting new case where this generalization applies occurs when A is a block algebra of G. In this case, the source algebra B is likely to contain all the ''local'' information about the block; for instance, the module categories of A and B are equivalent through restriction, and it is easy to see that vertices and sources of indecomposble A-modules can be computed from the corresponding B-modules; similarly, it is not difficult to show that the matrix of generalized decomposition numbers can be computed from B. Our main result here implies the analogous statement concerning the local category of the block where objects are the local pointed groups on A and morphisms are the G-exomorphisms, namely: the equivalence class of the local category can be computed from B. In order to prove that, we introduce the so called local fusion category of any interior G-algebra and we show that (i) local fusion categories of A and B are equivalent, (ii) local and local fusion categories of A coincide.
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 16S34
zbMATH DE Number: 3979341
zbMATH Keywords: vertices
zbMATH Keywords: sources
zbMATH Keywords: group rings
zbMATH Keywords: interior G-algebra
zbMATH Keywords: source algebras
zbMATH Keywords: block algebra
zbMATH Keywords: module categories
zbMATH Keywords: generalized decomposition numbers
zbMATH Keywords: local pointed groups
zbMATH Keywords: G-exomorphisms
zbMATH Keywords: local fusion category
MaRDI profile type: Publication
full work available at URL: 0021-8693(86)90221-8
OpenAlex ID: W2027218654
cites work: A Frobenius theorem for blocks
cites work: Some remarks on defect groups
cites work: Pointed groups and construction of characters
description: scientific article; zbMATH DE number 3979341