On the irreducibility of irreducible characters of simple Lie algebras

TL;DR

This paper proves that the characters of irreducible representations of simple Lie algebras, especially for SL(r), are irreducible after dividing out Weyl denominator factors, under certain coprimality conditions.

Contribution

It establishes a new irreducibility property of characters of irreducible representations of simple Lie algebras, including a specific result for SL(r) with coprimality conditions.

Findings

Characters are irreducible after dividing out Weyl denominator factors.

For SL(r), characters are strongly irreducible when certain weights are coprime.

The irreducibility holds for all powers of group elements under specified conditions.

Abstract

We establish an irreducibility property for the characters of finite dimensional, irreducible representations of simple Lie algebras (or simple algebraic groups) over the complex numbers, i.e., that the characters of irreducible representations are irreducible after dividing out by (generalized) Weyl denominator type factors. For $SL(r)$ the irreducibility result is the following: let $\lambda=(a_1\geq a_2\geq ... a_{r-1}\geq 0)$ be the highest weight of an irreducible rational representation $V_{\lambda}$ of $SL(r)$. Assume that the integers $a_1+r-1, ~a_2+r-2,..., a_{r-1}+1$ are relatively prime. Then the character $\chi_{\lambda}$ of $V_{\lambda}$ is strongly irreducible in the following sense: for any natural number $d$, the function $\chi_{\lambda}(g^d), ~g\in SL(r,\C)$ is irreducible in the ring of regular functions of $SL(r,\C)$.