The source algebras of nilpotent blocks over arbitrary ground-fields
type: Item
instance of: scholarly article
title: The source algebras of nilpotent blocks over arbitrary ground-fields
zbMATH Open document ID: 0815.20010
author: Yun Fan
published in: Journal of Algebra
publication date: 1995-07-04T00:00:00Z
review text: Let \(\mathcal O\) be a complete discrete valuation ring with residue field \(k\) of prime characteristic \(p\) that is not necessarily algebraically closed, let \(G\) be a finite group, set \(A = {\mathcal O} G\) and let \(b\) be a block idempotent of \(A\). Clearly \(A\) is an interior \(G\)-algebra. For any subgroup \(H\) of \(G\), a conjugacy class \(\beta\) of primitive idempotents of \(A^ H\) is said to be a point of \(H\) on \(A\) and the pair \(H_ \beta\) is called a pointed group on \(A\). For any \(i \in \beta\), \(iAi\) is a subalgebra of \(A\) and is unique up to an \(H\)-algebra embedding so set \(A_ \beta \equiv iAi\). For \(\alpha = \{b\}\), a point of \(G\) on \(A = {\mathcal O}G\), \(G_ \alpha\) is a pointed group on \(A\) and \(A_ \alpha = {\mathcal O}Gb\). If \(H_ \beta\), \(K_ \gamma\) are two pointed groups on \(A\), then \(K_ \gamma\) is contained in \(H_ \beta\) \((K_ \gamma \subset H_ \beta)\) if \(K \leq H\) and there are \(j \in \gamma\), \(i \in \beta\) such that \(ji = j = ij\). A pointed group \(Q_ \delta\) is said to be local if \(\text{Br}_ Q(\delta) \neq 0\) (hence \(Q\) is a \(p\)-subgroup). The maximal local pointed groups \(P_ \gamma\) contained in \(G_ \alpha\) (where \(\alpha = \{b\}\)) are called the defect pointed groups of \(G_ \alpha\) and are unique up to \(G\)-conjugation and the \(P\)-algebra \(A_ \gamma\)-is defined to be the source algebra of \(\alpha = \{b\}\). The local category of \(A_ \alpha\) as a \(G\)-algebra is defined by: objects are the local pointed groups on \(A_ \alpha\) and the set of morphisms \(E_ G(Q_ \delta,R_ \varepsilon)\) from \(Q_ \delta\) to \(R_ \varepsilon\) is the set \(\{g \in G \mid (Q_ \delta)^ g \subset R_ \varepsilon\) modulo \(\text{Inn}(R)\}\). A local pointed group \(Q_ \delta\) on \(A_ \gamma\) is also a local pointed group on \(A_ \alpha\) and \(E_ P(Q_ \delta,R_ \varepsilon) \subseteq E_ G(Q_ \delta,R_ \varepsilon)\). Thus there is an inclusion functor from the local category of \(A_ …
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 20D20
zbMATH DE Number: 589963
zbMATH Keywords: complete discrete valuation ring
zbMATH Keywords: finite groups
zbMATH Keywords: block idempotents
zbMATH Keywords: interior \(G\)-algebras
zbMATH Keywords: primitive idempotents
zbMATH Keywords: \(p\)-subgroups
zbMATH Keywords: maximal local pointed groups
zbMATH Keywords: defect pointed groups
zbMATH Keywords: local category
zbMATH Keywords: inclusion functor
zbMATH Keywords: source algebras
zbMATH Keywords: nilpotent blocks
zbMATH Keywords: irreducible Brauer character
zbMATH Keywords: Brauer categories
reviewed by: Morton E. Harris
MaRDI profile type: Publication
full work available at URL: jabr.1994.1134
OpenAlex ID: W2038350798
DOI: 10.1006/JABR.1994.1134
description: scientific article; zbMATH DE number 589963