Power sums of Coxeter exponents

TL;DR

This paper demonstrates that for any irreducible finite Coxeter system, the sum of the nth powers of Coxeter exponents can be expressed as a universal polynomial in four parameters, revealing a new uniform structure.

Contribution

It introduces a uniform polynomial formula for power sums of Coxeter exponents in terms of four parameters, generalizing previous specific cases.

Findings

Sum of powers expressed as a polynomial in four parameters

Provides a uniform formula applicable to all irreducible finite Coxeter systems

Enhances understanding of Coxeter exponents' algebraic structure

Abstract

Consider an irreducible finite Coxeter system. We show that for any nonnegative integer n the sum of the nth powers of the Coxeter exponents can be written uniformly as a polynomial in four parameters: h (the Coxeter number), r (the rank), and two further parameters.