Weyl Orbit Characters and Schur Functions

TL;DR

This paper introduces a method to compute characters of finite Lie algebras using Weyl orbit characters and Schur functions, simplifying calculations by avoiding Weyl group summations while maintaining consistency with the Weyl character formula.

Contribution

It demonstrates that for $A_N$ Lie algebras, characters can be derived from Weyl orbit characters and Schur functions, providing a more efficient computational approach.

Findings

Characters for $A_N$ Lie algebras are equivalent to Schur functions.

Weyl orbit characters can be used to define irreducible characters without summing over Weyl groups.

The method is compatible with the Weyl character formula.

Abstract

For finite Lie algebras, it is shown that characters can be defined first for Weyl orbits and then for irreducible representations. For $A_N$ Lie algebras, weight multiplicities can then be calculated by only stating that characters are equivalent to Schur functions. This also means that to calculate characters there is no need to sum over Weyl groups. The compatibility with the Weyl character formula will however be shown.