The characteristic polynomials of imprimitive groups and affine Coxeter groups

TL;DR

This paper investigates the characteristic polynomials of imprimitive groups and affine Coxeter groups, establishing their role in determining the characters of finite-dimensional representations.

Contribution

It introduces generators for defining characteristic polynomials of $G(r,1,n)$ and proves these polynomials determine the characters of all finite-dimensional representations, extending to affine Coxeter groups.

Findings

Characteristic polynomials determine representation characters.

Generators for characteristic polynomials of $G(r,1,n)$ are identified.

Results extend to affine Coxeter groups and $G(r,p,n)$.

Abstract

In this paper, we will seek appropriate generators to define the characteristic polynomials of $G(r,1,n)$, and prove that for every finite dimensional representation of $G(r,1,n)$, the characteristic polynomial of $G(r,1,n)$ determines the character of this representation. Furthermore, the same conclusion holds for affine Coxeter groups \(\widetilde{W}\) and $G(r,p,n)$.