#####
Séminaire Lotharingien de Combinatoire, B50b (2003), 27 pp.

# Alun O. Morris and Huw I. Jones

# Projective Representations
of Generalized Symmetric Groups

**Abstract.**
The representation theory of generalized symmetric groups has been
of interest over a long period dating back to the classical work
of W. Specht [28,29] and M.Osima
- an exposition of this work and other references may
be found in [12]. Furthermore, the projective
representations of these groups have been considered by a number
of authors, much of the this work was not published or was
published in journals not readily accessible in the western world.
The first comprehensive work on the projective representations of
the generalized symmetric groups was due to E. W. Read [24]
which was followed later by an improvement in the work of M.
Saeed-ul-Islam , see, for example, [26]. Of equal
interest has been the representation theory of the hyperoctahedral
groups, which are a special case of the generalized symmetric
groups. The projective representations of these groups was
considered by M. Munir in his thesis [20] which elaborated
on the earlier work of E. W. Read and M. Saeed-ul-Islam and also
by J. Stembridge [31] who gave an independent development
which was more complete and satisfactory in many respects. This
approach later influenced that used by H. I. Jones in his thesis
[13] where the use of Clifford algebras was emphasized.
More recently, the generalized symmetric groups have become far
more predominant in the context of complex reflection groups and
the corresponding cyclotomic Hecke algebras where they and their
subgroups form the infinite family *G*(*m,p,n*), see for example
[3], [4] and [5]. In view of this interest,
it was thought worthwhile to present this work which is based on
the earlier work of H. I. Jones which has not been published. As
this article is also meant to be partially expository, a great
deal of the background material is also presented.

Received: June 3, 2003.
Accepted: October 30, 2003.
Final Version: November 4, 2003.

The following versions are available: