On the Calculation of Group Characters

TL;DR

This paper presents a method to compute characters of irreducible representations of finite Lie algebras without using Weyl group summations, simplifying calculations especially for higher-rank algebras.

Contribution

It introduces a new approach to calculate Lie algebra characters directly from the Weyl formula, avoiding complex Weyl group summations.

Findings

Characters for G2 Lie algebra are computed without Weyl group sums.

The method simplifies calculations for higher-rank Lie algebras.

Technical details are provided for practical implementation.

Abstract

It is known that characters of irreducible representations of finite Lie algebras can be obtained using theWeyl character formula including Weyl group summations which make actual calculations almost impossible except for a few Lie algebras of lower rank. By starting from the Weyl character formula, we show that these characters can be re-expressed without referring to Weyl group summations. Some useful technical points are given in detail for the instructive example of G2 Lie algebra.