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Transactions of the American Mathematical Society

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Fayers' conjecture and the socles of cyclotomic Weyl modules


Authors: Jun Hu and Andrew Mathas
Journal: Trans. Amer. Math. Soc. 371 (2019), 1271-1307
MSC (2010): Primary 20G43, 20C08, 20C30, 05E10
DOI: https://doi.org/10.1090/tran/7551
Published electronically: September 10, 2018
MathSciNet review: 3885179
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Abstract: Gordon James proved that the socle of a Weyl module of a classical Schur algebra is a sum of simple modules labelled by $ p$-restricted partitions. We prove an analogue of this result in the very general setting of ``Schur pairs''. As an application we show that the socle of a Weyl module of a cyclotomic $ q$-Schur algebra is a sum of simple modules labelled by Kleshchev multipartitions and we use this result to prove a conjecture of Fayers that leads to an efficient LLT algorithm for the higher level cyclotomic Hecke algebras of type $ A$. Finally, we prove a cyclotomic analogue of the Carter-Lusztig theorem.


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Additional Information

Jun Hu
Affiliation: School of Mathematics, Beijing Institute of Technology, Beijing, 100081, People’s Republic of China
Email: junhu303@qq.com

Andrew Mathas
Affiliation: School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia
Email: andrew.mathas@sydney.edu.au

DOI: https://doi.org/10.1090/tran/7551
Keywords: Cyclotomic Hecke algebras, Schur algebras, quasi-hereditary and graded cellular algebras, Khovanov--Lauda--Rouquier algebras
Received by editor(s): February 21, 2016
Received by editor(s) in revised form: May 24, 2017
Published electronically: September 10, 2018
Additional Notes: Both authors were supported by the Australian Research Council. The first author was also supported by the National Natural Science Foundation of China (No. 11525102).
Article copyright: © Copyright 2018 American Mathematical Society