Kostant's Weight Multiplicity Formula and the Fibonacci and Lucas Numbers
Abstract
Consider the weight $\lambda$ which is the sum of all simple roots of a simple Lie algebra. Using Kostant's weight multiplicity formula we describe and enumerate the contributing terms to the multiplicity of the zero weight in the representation with highest weight $\lambda$. We prove that in Lie algebras of type $A$ and $B$, the number of contributing terms to the multiplicity of the zero-weight space in the representation with highest weight $\lambda$ is given by a Fibonacci number, and that in Lie algebras of type $C$ and $D$, the analogous result is given by a multiple of a Lucas number.
- Publication:
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arXiv e-prints
- Pub Date:
- November 2011
- DOI:
- 10.48550/arXiv.1111.6648
- arXiv:
- arXiv:1111.6648
- Bibcode:
- 2011arXiv1111.6648C
- Keywords:
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- Mathematics - Representation Theory;
- Mathematics - Combinatorics;
- 05E10
- E-Print:
- 11 pages