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

摘要

Consider the weight λ that is the sum of all simple roots of a simple Lie algebra g. Using Kostant's weight multiplicity formula we describe and enumerate the contributing terms to the multiplicity of an integral weight μ in the representation of g with highest weight λ, which we denote by L(λ). 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(λ) 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 μ 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 μ in L(λ), 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 μ in the representation L(λ) in all classical Lie algebras. Keywords and phrases: Kostant's weight multiplicity formula, Weyl alternation sets, combinatorial representation theory.