On Projective Characters of Rotation Sub- Group

Abstract

_ Let * be a root system in a £-dimensional real Euclidean space V with Weyl group W( £), and let W+{<f) denote its rotation subgroup. In [173 » the projective representations of the rotation subgroup W ( 9) have been determined from those of W( $) for each root system $. This is done by constructing non-trivial central extensions of W ( $) via the double coverings of the rotation groups SO(£). This adaptation gives a unified way of obtaining the basic projective representations of W+( $) from those of W($), determined in [9] . In particular, formulae giving irreducible characters of these representations are explicitly determined in each case. Our object here is to apply the fore-going results to Rotation subgroups of Weyl groups of types Dg and D-, that o is, those groups which have Schur multiplier (Z0) 0 Z_. f. o In particular, we give the a -regular classes for the factor set ex considered in [ 16] , as well as obtain basic projective characters for these groups. The following is a brief description of the individual chapters of this dissertation. In chapter 1, we give basic ideas of factor sets and projective representations of finite groups and some of the properties of these representations. In chapter 2, we present the concept of Schur multipliers and give the relationship between central extensions and projective representations of finite groups. Projective characters of finite groups and some of their properties, are given in chapter 3- Chapter 4 is mainly concerned with Weyl groups and their Rotation subgroups, and Schur

(ix) multipliers of these subgroups. The work in these chapters is applied in chapter 5, to obtain the basic projective characters of the Rotation subgroups of the Weyl groups of types D,- and D7. The results are summarized in Tables II and III.