Alun O. Morris

Decomposition numbers of exceptional Weyl groups, II

Abstract The decomposition numbers of the exceptional Weyl groups for the prime number p = 3 are obtained; other relevant prime numbers will be considered in forthcoming publications. General references on Weyl groups, the characters of the individual groups and on the modular representation theory required are given. The technique used for calculating the decomposition matrices are those described by James and Kerber together with the results of Brauer. Notation is introduced and then each individual group is considered and the results tabulated. For more details one may consult Khosraviyani, Ph.D. thesis, University College of Wales, May 1981.

Generalizations of the Cauchy and Schur identities

Abstract One of the generalizations proved is that ∑ p k(π) z(π) = p+n−1 p−1 − n−1 p−1 where the summation is over all partitions π = (1k1,…, nkn) of n with parts not divisible by p, k(π) = k1 + … + kn and z(π) = 1k1k1! … nknkn!. p = 1 gives the Cauchy identity and p = 2 the Schur identity. This identity is itself obtained as a particular case of a more general identity and the proof involves a generalization of certain symmetric functions, called Hall functions, which have played a major role in recent enumeration problems.

Fischer Matrices for Projective Representations of Generalized Symmetric Groups

The aim of this work is to calculate the Fischer matrices for the covering groups of the Weyl group of type Bn and the generalized symmetric group. It is shown that the Fischer matrices are the same as those in the ordinary case for the classes of Sn which correspond to partitions with all parts odd. For the classes of Sn which correspond to partitions in which no part is repeated more than m times, the Fischer matrices are shown to be different from the ordinary case.

Projective Representations of Finite Reflection Groups. III

Abstract The Schur multipliers of reflection groups were first constructed by I. Ihara and T. Yokonuma. Later, R. B. Howlett gave a more unified, but less concrete, construction for arbitrary Coxeter groups. In this paper, Clifford algebras are used to give a more unified approach than that originally used by I. Ihara and T. Yokonuma.