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

TL;DR

This paper investigates the dimensions of spaces spanned by symmetric powers of natural characters of generalized symmetric groups, providing bounds over complex numbers and prime characteristic fields.

Contribution

It extends previous methods to generalized symmetric groups, establishing new bounds on the dimensions of symmetric power spaces in various characteristics.

Findings

Derived upper and lower bounds for the dimension of symmetric power spaces.

Adapted existing methods to broader classes of groups.

Applicable over complex numbers and prime characteristic fields.

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.