Kostant’s weight multiplicity formula and the Fibonacci and Lucas numbers

Consider the weight $\lambda$ that is the sum of all simple roots of a simple Lie algebra $\mathfrak{g}$. Using Kostant’s weight multiplicity formula we describe and enumerate the contributing terms to the multiplicity of an integral weight \mu in the representation of $\mathfrak{g}$ with highest weight $\lambda$, which we denote by $L(\lambda)$. We prove that in Lie algebras of type $A$ and $B$, the number of terms contributing a nonzero value in the multiplicity of the zero-weight in $L(\lambda)$ is given by a Fibonacci number, and that in the Lie algebras of type $C$ and $D$, the analogous result is given by a multiple of a Lucas number. When $\mu$ is a nonzero integral weight we show that in Lie types $A$ and $B$ there is only one term contributing a nonzero value to the multiplicity of $\mu$ in $L(\lambda)$, and that in the Lie algebras of type $C$ and $D$, all terms contribute a value of zero. We conclude by using these results to compute the $q$-multiplicity of an integral weight $\mu$ in the representation $L(\lambda)$ in all classical Lie algebras.

Keywords

Kostant’s weight multiplicity formula, Weyl alternation sets, combinatorial representation theory

2010 Mathematics Subject Classification

05E10

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K. Chang was supported by a National Science Foundation grant (#DMS1148695) through the Center for Undergraduate Research in Mathematics (CURM) of Brigham Young University, and by corporate sponsors.

P. E. Harris was supported by National Science Foundation grant #DMS1620202.

Received 26 December 2017

Published 27 September 2019