The adjoint representation of a Lie algebra and the support of Kostant's weight multiplicity formula

TL;DR

This paper investigates the contributing terms in Kostant's weight multiplicity formula for the adjoint representation of classical Lie algebras, providing enumeration methods and explicit results for types B, C, D, and some non-zero weights.

Contribution

It introduces a systematic way to enumerate contributing terms in Kostant's formula for classical Lie algebras, including new results for non-zero weights and connections to Fibonacci numbers.

Findings

Enumeration of contributing terms for zero weight in types B, C, D

Cardinality of non-zero weight sets in the adjoint representation

Fibonacci number enumeration for certain non-zero weights

Abstract

Even though weight multiplicity formulas, such as Kostant's formula, exist their computational use is extremely cumbersome. In fact, even in cases when the multiplicity is well understood, the number of terms considered in Kostant's formula is factorial in the rank of the Lie algebra and the value of the partition function is unknown. In this paper we address the difficult question: What are the contributing terms to the multiplicity of the zero weight in the adjoint representation of a finite dimensional Lie algebra? We describe and enumerate the cardinalities of these sets (through linear homogeneous recurrence relations with constant coefficients) for the classical Lie algebras of Type $B$, $C$, and $D$, the Type $A$ case was computed by the first author in [5]. In addition, we compute the cardinality of the set of contributing terms for non-zero weight spaces in the adjoint representation. In the Type $B$ case, the cardinality of one such non-zero-weight is enumerated by the Fibonacci numbers. We end with a computational proof of a result of Kostant regarding the exponents of the respective Lie algebra for some low rank examples and provide a section with open problems in this area.