The complexity of certain Specht modules for the symmetric group

TL;DR

This paper investigates the complexity of Specht modules for symmetric groups, proving that certain modules have less than maximal complexity when their Young diagrams are composed of p-by-p blocks, confirming part of a conjecture.

Contribution

It introduces a new class of partitions related to p-weight and branching, proving they have less than maximal complexity, advancing understanding of Specht module complexities.

Findings

Specht modules with Young diagrams built from p-by-p blocks have less than maximal complexity.

The class of partitions arises naturally from questions about p-weight and branching.

The paper confirms one direction of a conjecture relating complexity to Young diagram structure.

Abstract

During the 2004-2005 academic year the VIGRE algebra research group at the University of Georgia computed the complexities of certain Specht modules S^\lambda for the symmetric group, using the computer algebra program Magma. The complexity of an indecomposable module does not exceed the p-rank of the defect group of its block. The Georgia group conjectured that, generically, the complexity of a Specht module attains this maximal value; that it is smaller precisely when the Young diagram of $\lambda$ is built out of $p \times p$ blocks. We prove one direction of this conjecture by showing these Specht modules do indeed have less than maximal complexity. It turns out that this class of partitions, which has not previously appeared in the literature, arises naturally as the solution to a question about the $p$-weight of partitions and branching.