On the determinant of representations of generalized symmetric groups ℤr≀Sn.

In this paper, we study the determinant of irreducible representations of the generalized symmetric groups Z r ≀ S n . We give an explicit formula to compute the determinant of an irreducible representation of Z r ≀ S n . Recently, several authors have characterized and counted the number of irreducible representations of a given finite group with nontrivial determinant. Motivated by these results, given an integer n, r an odd prime and ζ a nontrivial multiplicative character of Z r ≀ S n with n < r, we obtain an explicit formula to compute N ζ (n) , the number of irreducible representations of Z r ≀ S n whose determinant is ζ. [ABSTRACT FROM AUTHOR]