Overview

The generalized symmetric groups are the wreath product groups of the cyclic group with the symmetric group, a natural group-theoretic construction with many interesting applications. Some interesting special cases of these groups are the symmetric group and the hyperoctahedral group. We denote the wreath product groups (Z/rZ) ≀ Sn with G(n, r) throughout this thesis. The problem of counting the number of irreducible representations of a given group whose determinant is non-trivial gains interest for re- searchers recently. In the case of symmetric groups, they call such representations to be chiral if the composition of ρ with the determinant map is non-trivial. The problem of counting the non-trivial determinants in [7] and [13] have their genesis in [28]. In [28], Macdonald developed combinatorics for partitions and gave a closed formula to count the number of odd-dimensional Specht modules for the symmetric groups. This number happened to be the product of the powers of 2 in the binary expansion of n and was obtained by characterizing the 2-core tower of the odd partitions.

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