Vertices of Specht modules and blocks of the symmetric group

TL;DR

This paper investigates the vertices of Specht modules for symmetric groups, providing new methods to determine defect groups and Green correspondents, and offering a novel proof of the Brauer correspondence for symmetric group blocks.

Contribution

It introduces a main theorem linking Specht module vertices to large subgroups, enabling new approaches to defect groups and Green correspondents in symmetric groups.

Findings

Identifies large subgroups contained in vertices of Specht modules.

Provides a new method to determine defect groups of symmetric groups.

Offers a new proof of the Brauer correspondence for symmetric group blocks.

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. We also use it to find the Green correspondents of a particular family of simple Specht modules; as a corollary, we get 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. Brou{\'e}, together with combinatorial arguments using Young tableaux.