Unitriangular basic sets for blocks of the symmetric and alternating groups of small weights

TL;DR

This paper investigates the existence of special unitriangular basic sets for symmetric group blocks of small weights, providing a natural labeling for modular irreducible representations and analyzing their combinatorial structure.

Contribution

It establishes the existence of stable unitriangular basic sets for p-blocks of weight 2 in symmetric groups for any odd prime p, with a detailed combinatorial description.

Findings

Existence of stable unitriangular basic sets for weight 2 blocks

Description of these sets via partition combinatorics

Behavior with respect to the Mullineux involution

Abstract

We study the existence of unitriangular basic sets for the symmetric group which behave nicely with respect to the Mullineux involution. Such sets give a natural labelling for the modular irreducible representations. We show that, for any odd prime p, the p-blocks of the symmetric group with weight 2 have stable unitriangular basic sets which we describe by studying the combinatorics of partitions in these blocks.