Characteristic and Coxeter polynomials for affine Lie algebras

TL;DR

This paper computes characteristic and Coxeter polynomials for affine Lie algebras using Chebyshev polynomials, providing explicit factorizations and methods to determine exponents and Coxeter numbers.

Contribution

It introduces new explicit factorizations of these polynomials and methods for calculating exponents and Coxeter numbers for affine Lie algebras.

Findings

Explicit factorizations of polynomials as cyclotomic products

Methods for obtaining exponents and Coxeter numbers

Computed exponents and Coxeter numbers for $A_n^{(1)}$

Abstract

We compute the characteristic polynomials of affine Cartan, adjacency matrices and Coxeter polynomials of the associated Coxeter system using Chebyshev polynomials. We give explicit factorization of these polynomials as products of cyclotomic polynomials. Finally, we present several different methods of obtaining the exponents and Coxeter number for affine Lie algebras. In particular we compute the exponents and Coxeter number for each conjugacy class in the case of $A_n^{(1)}$.