Pointed groups and construction of modules
type: Item
instance of: scholarly article
title: Pointed groups and construction of modules
zbMATH Open document ID: 0658.20004
DOI: 10.1016/0021-8693(88)90195-0
author: Lluis Puig
published in: Journal of Algebra
publication date: 1988-01-01T00:00:00Z
review text: Let \({\mathcal O}\) be a complete discrete valuation ring with residue field k of characteristic p. Let A be an \({\mathcal O}\)-algebra, finitely generated and free as an \({\mathcal O}\)-module. Let \(A^*\) be the group of invertible elements of A. Let G be a finite group. A is an interior G-algebra if there is a group homomorphism \(G\to A^*\) and G acts by inner automorphisms, giving A an \({\mathcal O}G\)-module structure. For an \({\mathcal O}G\)-module M and a subgroup H (\(\leq G)\), \(M^ H\) is the \({\mathcal O}H\)- submodule of M of those elements fixed by H. \({\mathcal P}(A)\) is the set of \(A^*\)-conjugacy classes of primitive idempotents of A. A pointed group \(H_{\beta}\) on A consists of a subgroup H together with \(\beta\in {\mathcal P}(A^ H)\); \(\beta\) is a point of H on A. For instance, if \(A={\mathcal O}G\), then \(A^ G\) is the centre of \({\mathcal O}G\) and \(\beta\) is a set containing one primitive central idempotent of \({\mathcal O}G\) (i.e. a block). Again for an \({\mathcal O}G\)-module M, one has an interior G- algebra structure on \(A=End_{{\mathcal O}}(M)\); \(A^ G=End_{{\mathcal O}G}(M)\) and a primitive idempotent \(i\in \beta \in p(A^ G)\) is the projection endomorphism onto an \({\mathcal O}G\)-indecomposable direct summand of \(M.\) P\({}_{\gamma}\) is called local if \(i\in \gamma\) does not lie in the sum of images of the trace functions from various \(A^ K\) for proper subgroups K of P; then P is necessarily a p-subgroup of G. Pointed subgroups are partially ordered by inclusion of subgroups and ``inclusion of idempotents''. A maximal local pointed subgroup of \(H_{\beta}\) is called a defect pointed group of \(H_{\beta}\); then H acts transitively on the set of defect pointed groups of \(H_{\beta}\). When \(A={\mathcal O}G\), a defect pointed group \(P_{\gamma}\) of the block \(H_{\beta}\) corresponds to the defect group of the block. When \(A=End_{{\mathcal O}}(M)\), the defect pointed group \(P_{\gamma}\) of the indec…
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20C05
zbMATH DE Number: 4075370
zbMATH Keywords: interior G-algebra
zbMATH Keywords: primitive idempotents
zbMATH Keywords: pointed group
zbMATH Keywords: trace functions
zbMATH Keywords: maximal local pointed subgroup
zbMATH Keywords: defect pointed groups
zbMATH Keywords: defect group
zbMATH Keywords: Green ring
zbMATH Keywords: residual Green ring
zbMATH Keywords: virtual module
zbMATH Keywords: Brauer's Second Main Theorem
zbMATH Keywords: blocks
zbMATH Keywords: generalized decomposition numbers
zbMATH Keywords: modular characters
zbMATH Keywords: virtual character
MaRDI profile type: Publication
full work available at URL: 0021-8693(88)90195-0
OpenAlex ID: W2092337834
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description: scientific article; zbMATH DE number 4075370