Fayers' conjecture and the socles of cyclotomic Weyl modules

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.

Latex, 26 pages. Revision that, in particular, fixes an issue with the "cyclotomic Carter-Lusztig theorem"

References in corpus (3)

• Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras
• Blocks of cyclotomic Hecke algebras
• Quiver Schur algebras and Koszul duality

Cited by in corpus (1)

• Content systems and deformations of cyclotomic KLR algebras of type $A$ and $C$