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

Let p p be an odd prime, and A n \mathfrak {A}_n the alternating group of degree n n . We determine which ordinary irreducible representations of A n \mathfrak {A}_n remain irreducible in characteristic p p , verifying the author’s conjecture from 2010. Given the preparatory work done in 2010, our t...

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Published in:Transactions of the American Mathematical Society 2016-08, Vol.368 (8), p.5807-5855
Main Author: Fayers, Matthew
Format: Article
Language:English
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Summary:Let p p be an odd prime, and A n \mathfrak {A}_n the alternating group of degree n n . We determine which ordinary irreducible representations of A n \mathfrak {A}_n remain irreducible in characteristic p 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 p having exactly two composition factors. This is accomplished through the use of the Robinson–Brundan–Kleshchev ‘ i 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.
ISSN:0002-9947
1088-6850
DOI:10.1090/tran/6531