On the principal blocks of \({\mathfrak S}_9\) and \({\mathfrak S}_{10}\) over a field of characteristic \(3\)
type: Item
instance of: scholarly article
title: On the principal blocks of \({\mathfrak S}_9\) and \({\mathfrak S}_{10}\) over a field of characteristic \(3\)
zbMATH Open document ID: 0986.20013
DOI: 10.1016/S0022-4049(99)00082-1
author: Kai Meng Tan
published in: Journal of Pure and Applied Algebra
publication date: 2002-02-04T00:00:00Z
review text: Following the work of \textit{S. Martin} and \textit{L. Russell} [J. Algebra 185, No. 2, 440-480 (1996; Zbl 0894.20016)], the author constructs the Ext-quiver of the principal block of the group algebra \(F{\mathfrak S}_9\) over an algebraically closed field \(F\) of characteristic 3. He finds in particular that there are no self-extensions of simple modules, the \(\text{Ext}^1\)-space between two simple modules is at most one-dimensional, and the Ext-quiver is bipartite. He further computes the complete Loewy structures of the principal indecomposable modules finding that they all have Loewy length 7. Using the results for \(F{\mathfrak S}_9\), he constructs the Ext-quiver of the principal block of \(F{\mathfrak S}_{10}\). The same extension results hold and he also determines the Loewy structures of the principal indecomposable modules. Again they all have Loewy length 7. In addition to their intrinsic value, these results are of interest because these blocks have non-Abelian defect groups. Whereas, previous investigations have primarily focused on blocks with Abelian defect group.
Mathematics Subject Classification ID: 20C30
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 20J06
zbMATH DE Number: 1526264
zbMATH Keywords: symmetric groups
zbMATH Keywords: Ext-quivers
zbMATH Keywords: principal blocks
zbMATH Keywords: principal indecomposable modules
zbMATH Keywords: non-Abelian defect groups
zbMATH Keywords: group algebras
zbMATH Keywords: simple modules
zbMATH Keywords: Loewy lengths
reviewed by: Christopher P. Bendel
MaRDI profile type: Publication
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full work available at URL: s0022-4049(99)00082-1
OpenAlex ID: W2055967160
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description: scientific article; zbMATH DE number 1526264