Fayers' conjecture and the socles of cyclotomic Weyl modules

TL;DR

This paper extends classical results about socles of Weyl modules to cyclotomic $q$-Schur algebras, proving Fayers' conjecture and providing tools for efficient algorithms in higher level cyclotomic Hecke algebras.

Contribution

It generalizes James' classical result to a broad setting of Schur pairs and proves Fayers' conjecture, linking socles to Kleshchev multipartitions and algorithms.

Findings

Socle of cyclotomic Weyl modules is a sum of simple modules labeled by Kleshchev multipartitions.

Proves Fayers' conjecture relating to socles and partitions.

Establishes a cyclotomic analogue of the Carter-Lusztig theorem.

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.