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Vertices of Specht modules and blocks of the symmetric group

Authors
Journal
Journal of Algebra
0021-8693
Publisher
Elsevier
Publication Date
Volume
323
Issue
8
Identifiers
DOI: 10.1016/j.jalgebra.2010.01.014
Keywords
  • Vertex
  • Specht Module
  • Symmetric Group
  • Block Theory
  • Defect Group
Disciplines
  • Mathematics

Abstract

Abstract This paper studies the vertices, in the sense defined by J.A. Green, of Specht modules for symmetric groups. The main theorem gives, for each indecomposable non-projective Specht module, a large subgroup contained in one of its vertices. A corollary of this theorem is a new way to determine the defect groups of symmetric groups. The main theorem is also used to find the Green correspondents of a particular family of simple Specht modules; as a corollary, this gives a new proof of the Brauer correspondence for blocks of the symmetric group. The proof of the main theorem uses the Brauer homomorphism on modules, as developed by M. Broué, together with combinatorial arguments using Young tableaux.

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