Thomas Gerber

Generalised canonical basic sets for Ariki-Koike algebras

Abstract Let H be a non-semi-simple Ariki–Koike algebra. According to [20] and [16] , there is a generalisation of Lusztigs a-function which induces a natural combinatorial order (parametrised by a tuple m) on Specht modules. In some cases, Geck and Jacon have proved that this order makes the decomposition matrix of H unitriangular. The algebra H is then said to admit a “canonical basic set”. We fully classify which values of m yield a canonical basic set for H and which do not. When this is the case, we describe these sets in terms of “twisted Uglov” or “twisted Kleshchev” multipartitions.

Branching graphs for finite unitary groups in nondefining characteristic

ABSTRACT We show that the modular branching rule (in the sense of Harish-Chandra) on unipotent modules for finite unitary groups is piecewise described by particular connected components of the crystal graph of well-chosen Fock spaces, under favorable conditions. Besides, we give the combinatorial formula to pass from one to the other in the case of modules arising from cuspidal modules of defect 0. This partly proves a recent conjecture of Jacon and the authors.