Characteristic polynomials for classical Lie algebras

TL;DR

This paper computes characteristic polynomials for finite-dimensional representations of classical and G2 Lie algebras, revealing their decomposition into Weyl group-invariant orbital factors.

Contribution

It introduces a method to explicitly compute and decompose characteristic polynomials using Weyl group orbits and invariant polynomial theory.

Findings

Characteristic polynomials decompose into irreducible orbital factors.

Decomposition is invariant under Weyl group actions.

Provides explicit formulas for classical and G2 Lie algebras.

Abstract

In this paper, we will compute the characteristic polynomials for finite dimensional representations of classical complex Lie algebras and the exceptional Lie algebra of type G2, which can be obtained through the orbits of integral weights under the action of their corresponding Weyl groups and the invariant polynomial theory of the Weyl groups. We show that the characteristic polynomials can be decomposed into products of irreducible orbital factors, each of which is invariant under the action of their corresponding Weyl groups.