Irreducible projective representations of the symmetric group which remain irreducible in characteristic $2$

TL;DR

This paper investigates which irreducible representations of double covers of symmetric groups remain irreducible in characteristic 2, using decomposition matrices, modular branching rules, and dimension comparisons.

Contribution

It provides a classification of irreducible projective representations of symmetric groups that stay irreducible in characteristic 2, introducing new techniques for analyzing decomposition matrices and branching rules.

Findings

Identifies irreducible projective representations remaining irreducible in characteristic 2

Constructs part of the decomposition matrix for a Rouquier block

Utilizes Brundan–Kleshchev modular branching rules and dimension comparisons

Abstract

For any finite group $G$ and any prime $p$ one can ask which ordinary irreducible representations remain irreducible in characteristic $p$. We answer this question for $p=2$ when $G$ is a proper double cover of the symmetric group. Our techniques involve constructing part of the decomposition matrix for a Rouquier block of a double cover, restricting to subgroups using the Brundan--Kleshchev modular branching rules and comparing the dimensions of irreducible representations via the bar-length formula.