Vertices of Specht modules and blocks of the symmetric group

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.