The Action of Generalised Symmetric Groups on Symmetric and Exterior Powers of Their Natural Representations

Abstract

We establish upper and lower bounds on the dimension of the space spanned by the symmetric powers of the natural character of generalised symmetric groups. We adapt the methods of Savitt and Stanley from their paper `A note on the symmetric powers of the standard representation of $S_n$' (The Electronic Journal of Combinatorics 7 (2000), R6) to obtain bounds both over the complex numbers and in prime characteristic.