On character generators for simple Lie algebras

TL;DR

This paper presents new formulas and methods for calculating character generating functions of simple Lie algebras, clarifying their structure and simplifying their computation, especially for rank-two cases.

Contribution

It introduces a new general formula for character generators, clarifies the roles of basis elements, and extends previous Demazure-based results for simple Lie algebras.

Findings

New general formula for character generators

Simplified construction of fundamental generalized-poset graphs

Detailed analysis of rank-two Lie algebras

Abstract

We study character generating functions (character generators) of simple Lie algebras. The expression due to Patera and Sharp, derived from the Weyl character formula, is first reviewed. A new general formula is then found. It makes clear the distinct roles of ``outside'' and ``inside'' elements of the integrity basis, and helps determine their quadratic incompatibilities. We review, analyze and extend the results obtained by Gaskell using the Demazure character formulas. We find that the fundamental generalized-poset graphs underlying the character generators can be deduced from such calculations. These graphs, introduced by Baclawski and Towber, can be simplified for the purposes of constructing the character generator. The generating functions can be written easily using the simplified versions, and associated Demazure expressions. The rank-two algebras are treated in detail, but we believe our results are indicative of those for general simple Lie algebras.