Layer structure of irreducible Lie algebra modules

TL;DR

This paper introduces the concept of layer sums for irreducible modules of simple Lie algebras, providing a new method for character decomposition, weight multiplicity computation, and a polynomial formula for counting weights.

Contribution

It presents a novel layer sum decomposition of characters, offering new computational techniques and a closed-form polynomial for counting weights in irreducible Lie algebra modules.

Findings

Layer sums decompose characters with non-negative integer coefficients.

New approach simplifies computation of Weyl characters and weight multiplicities.

Number of distinct weights is given by a polynomial in Dynkin labels, degree equal to the algebra's rank.

Abstract

Let $\mathfrak{g}$ be a finite-dimensional simple complex Lie algebra. A layer sum is introduced as the sum of formal exponentials of the distinct weights appearing in an irreducible $\mathfrak{g}$-module. It is argued that the character of every finite-dimensional irreducible $\mathfrak{g}$-module admits a decomposition in terms of layer sums, with only non-negative integer coefficients. Ensuing results include a new approach to the computation of Weyl characters and weight multiplicities, and a closed-form expression for the number of distinct weights in a finite-dimensional irreducible $\mathfrak{g}$-module. The latter is given by a polynomial in the Dynkin labels, of degree equal to the rank of $\mathfrak{g}$.