Parabolic induction for Hecke algebras

• Parabolische Induktion für Heckealgebren

Schönnenbeck, Christoph; Hiß, Gerhard (Thesis advisor); Geck, Meinolf (Thesis advisor); Fourier, Ghislain Paul Thomas (Thesis advisor)

Aachen (2019)
Dissertation / PhD Thesis

Dissertation, RWTH Aachen University, 2019

Abstract

The induction of representations of a subgroup W_0 to an overgroup W is one of the fundamental concepts in representation theory. If W is a Coxeter group, then one is particularly interested in the case where W_0 is a so-called parabolic subgroup of W, which implies that W_0 , too, is a Coxeter group. The corresponding induction is called parabolic induction. In this thesis we consider generalisations of this setup. We study the structure of parabolically induced modules of finite reflection groups and their Hecke algebras, more precisely of Coxeter groups, complex reflection groups, Iwahori-Hecke algebras, and cyclotomic Hecke algebras. We begin by recalling general concepts and results on the representation theory of algebras before introducing the various groups and algebras on which we work throughout. Then we show for almost all classes of reflection groups and Hecke algebras considered in this thesis that parabolic induction from proper parabolic subgroups or subalgebras always yields reducible modules, i.e. no irreducible modules are obtained from parabolic induction. Next we turn to parabolic induction of Ariki-Koike algebras, a subclass of cyclotomic Hecke algebras. Here, so-called categorification results enable us under certain circumstances to make use of combinatorial arguments in terms of directed graphs known as crystal graphs. We explain the necessary combinatorics of multipartitions and crystals graphs and use this to prove a lower bound on the number of irreducible constituents of parabolically induced modules of Ariki-Koike algebras. Moreover, we give an analogous bound in the cases where the combinatorial arguments are not applicable. We apply our results on Ariki-Koike algebras to also study parabolic induction for the closely related cyclotomic rational Cherednik algebras. In the second half of this thesis we investigate the connection between induction and specialisation, i.e. a change of the base ring of an algebra. We do this first in a general setting and then more specifically in the context of parabolic induction for Hecke algebras. In particular, we show how the connection between parabolic induction and specialisation allows us to apply results from the ordinary representation theory of groups and so-called decomposition numbers to compute the irreducible constituents of parabolically induced modules of Iwahori-Hecke algebras. Moreover, we explain how in some cases parabolic induction can be studied with the help of specialisation and Clifford theory. Finally, we give a description of the computational methods we used to obtain the decomposition numbers of exceptional Iwahori-Hecke algebras in so-called bad characteristic.

Institutions

• Department of Mathematics [110000]
• Chair of Algebra and Number Theory [114710]