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Kostant’s weight multiplicity formula and the Fibonacci and Lucas numbers. (English) Zbl 1428.17004
Summary: 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.

MSC:
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
17B20 Simple, semisimple, reductive (super)algebras
05E10 Combinatorial aspects of representation theory
11B39 Fibonacci and Lucas numbers and polynomials and generalizations
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