Defect \(3\) blocks of symmetric group algebras
type: Item
instance of: scholarly article
title: Defect \(3\) blocks of symmetric group algebras
zbMATH Open document ID: 0947.20011
author: S. Martin
author: Lee Russell
published in: Journal of Algebra
publication date: 2000-11-05T00:00:00Z
review text: Let \(k\) be a field of characteristic \(p\geq 5\) and let \(S_n\) denote the symmetric group of degree \(n\geq 1\). In a previous work, the authors studied the principal block \(B_0(kS_{3p})\), which is the unique block of \(kS_{3p}\) of defect \(3\). In particular they showed that, in \(B_0(kS_{3p})\), all decomposition numbers are \(0\) or \(1\), there are no self-extensions of irreducibles and that all extensions between irreducibles are of dimension at most \(1\). In this paper, they show that these properties hold for all defect \(3\) blocks of symmetric group algebras. In fact, their main theorem is: Theorem 7.1. Let \(p\geq 5\). A block \(B\) of \(kS_n\) of defect \(3\) has the following properties: (1) All of the decompositions are \(0\) or \(1\). (2) \((D^\lambda_k,D^\lambda_k)^1_{kS_n}=0\) for all \(p\)-regular \(\lambda\). (3) \((D^\lambda_k,D^\mu_k)=0\) or \(1\) for all \(p\)-regular \(\lambda\) and \(\mu\).
reviewed by: Morton E. Harris
Mathematics Subject Classification ID: 20C30
Mathematics Subject Classification ID: 20C05
zbMATH DE Number: 1270884
zbMATH Keywords: symmetric groups
zbMATH Keywords: principal blocks
zbMATH Keywords: decomposition numbers
zbMATH Keywords: defect \(3\) blocks
zbMATH Keywords: symmetric group algebras
MaRDI profile type: Publication
full work available at URL: jabr.1998.7649
OpenAlex ID: W2038426969
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DOI: 10.1006/JABR.1998.7649
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description: scientific article; zbMATH DE number 1270884