Irreducible spin representations of symmetric and alternating groups which remain irreducible in characteristic 3

TL;DR

This paper classifies which irreducible spin representations of double covers of symmetric and alternating groups remain irreducible in characteristic 3, advancing understanding of modular representation theory for these groups.

Contribution

It provides a classification of irreducible spin representations in characteristic 3 for double covers of symmetric and alternating groups, except for a specific family related to spin RoCK blocks.

Findings

Classification achieved for most cases, excluding certain spin RoCK block families.

Techniques include induction, restriction, degree calculations, and decomposition of projective characters.

Utilizes recent results on spin RoCK blocks by Kleshchev and Livesey.

Abstract

For any finite group $G$ and any prime $p$ one can ask which ordinary irreducible representations remain irreducible in characteristic $p$, or more generally, which representations remain homogeneous in characteristic $p$. In this paper we address this question for $p=3$ when $G$ is a proper double cover of the symmetric or alternating group. We obtain a classification except in the case of a certain family of partitions relating to spin RoCK blocks. Our techniques involve induction and restriction, degree calculations, decomposing projective characters and recent results of Kleshchev and Livesey on spin RoCK blocks.