Simple modules of cyclotomic Hecke algebras

Abstract

Ariki showed that the simple modules of the cyclotomic Hecke algebra are labelled by Kleshchev multipartitions. Recently, Jacon gave an alternative recursive description of Uglov multipartitions, which can be thought of as a generalisation of Kleshchev multipartitions. In this thesis we extend Jacon's combinatorics and then give a non-recursive description of Kleshchev multipartitions. We then use these combinatorial tools in the framework of the diagramatic Cherednik algebras to give a complete classification of the simple modules coming from the Webster-Bowman "many cellular bases" indexed by a loading. In particular, we recover Ariki's classification theorem in the case of Kleshchev multipartitions. As a consequence we also obtain a new lower bound for the graded dimensions of the simple modules.