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


Author: Matthew Fayers
Journal: Trans. Amer. Math. Soc. 368 (2016), 5807-5855
MSC (2010): Primary 20C30, 05E10, 20C20
DOI: https://doi.org/10.1090/tran/6531
Published electronically: December 3, 2015
MathSciNet review: 3458400
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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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Additional Information

Matthew Fayers
Affiliation: School of Mathematical Sciences, Queen Mary University of London, Mile End Road, London E1 4NS, United Kingdom
Email: m.fayers@qmul.ac.uk

DOI: https://doi.org/10.1090/tran/6531
Received by editor(s): November 27, 2013
Received by editor(s) in revised form: July 25, 2014
Published electronically: December 3, 2015
Article copyright: © Copyright 2015 American Mathematical Society