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

Fayers, Matthew

doi:10.1090/tran/6531

The irreducible representations of the alternating group which remain irreducible in characteristic $p$
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Trans. Amer. Math. Soc. 368 (2016), 5807-5855 Request permission

Abstract:

Let $p$ be an odd prime, and $\mathfrak {A}_n$ the alternating group of degree $n$. We determine which ordinary irreducible representations of $\mathfrak {A}_n$ 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.

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