Irreducible characters of the generalized symmetric group

TL;DR

This paper develops new iterative algorithms for computing irreducible characters of the generalized symmetric group, reestablishes a key rule via algebraic methods, and explores relationships with modular characters.

Contribution

It introduces a novel iterative formula for characters of the generalized symmetric group and re-proves the Murnaghan-Nakayama rule using vertex algebraic techniques.

Findings

Derived a new iterative formula for generalized symmetric group characters

Reproved the Ariki-Koike version of the Murnaghan-Nakayama rule

Discovered a numerical relation between group characters and modular characters

Abstract

The paper studies how to compute irreducible characters of the generalized symmetric group $C_k\wr{S}_n$ by iterative algorithms. After reproving the Ariki-Koike version of the Murnaghan-Nakayama rule by vertex algebraic methods, we formulate a new iterative formula for characters of the generalized symmetric group. As applications, we find a numerical relation between the character values of $C_k\wr S_n$ and modular characters of $S_{kn}$.