Generating functions and multiplicity formulas: the case of rank two simple Lie algebras

TL;DR

This paper introduces a method using generating functions of characters to derive explicit weight multiplicity formulas for all representations of rank two simple Lie algebras, enhancing understanding of their structure.

Contribution

It presents a novel procedure leveraging generating functions to compute weight multiplicities across all representations of rank two simple Lie algebras.

Findings

Derived explicit multiplicity formulas for rank two simple Lie algebras

Provided a systematic method to extract weight multiplicities from generating functions

Enhanced computational tools for studying Lie algebra representations

Abstract

A procedure is described that makes use of the generating function of characters to obtain a new generating function $H$ giving the multiplicities of each weight in all the representations of a simple Lie algebra. The way to extract from $H$ explicit multiplicity formulas for particular weights is explained and the results corresponding to rank two simple Lie algebras shown.