## 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.

##### Authors

Brown, Benjamin Charles##### Collections

- Theses [3711]