Fayers’ conjecture and the socles of cyclotomic Weyl modules

Hu, Jun; Mathas, Andrew

doi:10.1090/tran/7551

Fayers’ conjecture and the socles of cyclotomic Weyl modules
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by Jun Hu and Andrew Mathas PDF

Trans. Amer. Math. Soc. 371 (2019), 1271-1307 Request permission

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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