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

Let p be an odd prime, and U n the alternating group of degree n. We determine which ordinary irreducible representations of U n remain irreducible in characteristic p, verifying the author's conjecture from 2010. Given the preparatory work done in 2010, 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. 2010 Mathematics Subject Classification. Primary 20C30, 05E10, 20C20.