Journal of Combinatorics

Volume 7 (2016)

Number 1

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

Pages: 75 – 116

DOI: https://dx.doi.org/10.4310/JOC.2016.v7.n1.a4

Authors

Pamela E. Harris (Department of Mathematical Sciences, United States Military Academy, West Point, New York, U.S.A.)

Erik Insko (Department of Mathematics, Florida Gulf Coast University, South Fort Myers, Florida, U.S.A.)

Lauren Kelly Williams (Department of Mathematics and Computer Systems, Mercyhurst University, Erie, Pennsylvania, U.S.A.)

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 classical Lie algebra? We describe and enumerate the cardinalities of these sets (through linear homogeneous recurrence relations with constant coefficients) for the classical Lie algebras $\mathfrak{so}_{2r+1} (\mathbb{C})$, $\mathfrak{sp}_{2r} (C)$, and $\mathfrak{so}_{2r} (C)$. The $\mathfrak{sl}_{r+1} (C)$ case was computed by the first author in [7]. In addition, we compute the cardinality of the set of contributing terms for non-zero weight spaces in the adjoint representation. In the $\mathfrak{so}_{2r+1} (C)$ case, the cardinality of one such non-zero weight is enumerated by the Fibonacci numbers.

Keywords

Kostant’s weight multiplicity formula, adjoint representation, combinatorial representation theory

2010 Mathematics Subject Classification

05E10

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Published 9 December 2015