Some Problems in the Representation Theory of Hyperoctahedral Groups and Related Algebras

Abstract

We begin by using a version of Green correspondence due to Grabmeier to count the number of components of two permutation modules V®r and y®r for the hyperoctahedral group. We quantize these actions to make v®r and y®r into modules for the type B Hecke algebra 1i(r) and then show that, as 1i(r)-modules, y®r is isomorphic to a direct sum of permutation modules MA as given by Du and Scott. This enables us to use our earlier results to show that in the group case, over odd characteristic, the q-Schur2 algebra and the hyperoctahedral Schur algebra are Morita equivalent, as these algebras are respectively the centralizing algebras of the actions of the hyperoctahedral group on y®r and v®r. We then attempt to construct a bialgebra, the dual of whose rth homogeneous part is isomorphic to the q-Schur2 algebra. We show that this is not possible by the usual methods unless q = 1, and give a full description in the group case. Results of earlier chapters lead us to introduce the notion of a balanced Mackey system for a finite group G, and exhibit balanced Mackey systems for wreath products of H and the symmetric group, where H is any finite group, and a new balanced Mackey system for the symmetric group itself. We then use this as a basis for counting the number of simple modules for the partition algebra, and also derive a formula for the dimensions of these simple modules. In the final chapter we conjecture how some of our results may extend to complex reflection groups and Ariki-Koike algebras.