The complexities of some simple modules of the symmetric groups

TL;DR

This paper investigates the complexities of simple modules in Rouquier blocks of symmetric groups, revealing that their complexity equals the p-weight when less than p, and providing specific complexity values for certain modules.

Contribution

It establishes the common complexity of simple modules in Rouquier blocks with p-weight less than p and determines the complexities of specific modules when p is odd.

Findings

Simple modules in Rouquier blocks with p-weight w < p have complexity w.

When p is odd, D^{(p+1,1^{p-1})} has complexity 1.

Other simple modules with p-weight 2 have complexity 2.

Abstract

We show that the simple modules of the Rouquier blocks of symmetric groups, in characteristic $p$ and having $p$-weight $w$ with $w < p$, have a common complexity $w$, and that when $p$ is odd, $D^{(p+1,1^{p-1})}$ has complexity 1, while the other simple modules labelled by a partition having $p$-weight 2 have complexity 2.