The irreducible representations of the alternating group which remain irreducible in characteristic p

TL;DR

This paper classifies which irreducible representations of the alternating group remain irreducible over a field of characteristic p, confirming a previous conjecture and using advanced combinatorial and algebraic tools.

Contribution

It verifies a conjecture by determining the irreducibility of representations of A_n in characteristic p using Specht modules and i-restriction functors.

Findings

Identifies which irreducible A_n representations stay irreducible in characteristic p.

Uses i-restriction functors and known decomposition results.

Confirms the author's conjecture from previous work.

Abstract

Let p be an odd prime, and A_n the alternating group of degree n. We determine which ordinary irreducible representations of A_n remain irreducible in characteristic p, verifying the author's conjecture from [Represent. Theory 14, 601-626]. Given the preparatory work done in [op. cit.], our task is to determine which self-conjugate partitions label Specht modules for the symmetric group in characteristic p having exactly two composition factors. This is accomplished through the use of the Robinson-Brundan-Kleshchev 'i-restriction' functors, together with known results on decomposition numbers for the symmetric group and additional results on the Mullineux map and homomorphisms between Specht modules.