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.

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