×

zbMATH — the first resource for mathematics

On the adjoint representation of and the Fibonacci numbers. (English. French summary) Zbl 1273.17010
Summary: We decompose the adjoint representation of \(\mathfrak{sl}_{r+1} = \mathfrak{sl}_{r+1}(\mathbb C)\) by a purely combinatorial approach based on the introduction of a certain subset of the Weyl group called the Weyl alternation set associated to a pair of dominant integral weights. The cardinality of the Weyl alternation set associated to the highest root and zero weight of \(\mathfrak{sl}_{r+1}\) is given by the \(r\)th Fibonacci number. We then obtain the exponents of \(\mathfrak{sl}_{r+1}\) from this point of view.

MSC:
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
11B39 Fibonacci and Lucas numbers and polynomials and generalizations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] C. Cochet, Vector partition function and representation theory, Conference Proceedings Formal Power Series and Algebraic Combinatorics, Taormina, Sicile, 2005, 12 pages. · Zbl 1078.22006
[2] Goodman, R.; Wallach, N.R., Symmetry, representations and invariants, (2009), Springer New York · Zbl 1173.22001
[3] Humphreys, J.E., Reflection groups and Coxeter groups, (1990), Cambridge University Press Cambridge · Zbl 0725.20028
[4] Kostant, B., A formula for the multiplicity of a weight, Proc. natl. acad. sci. USA, 44, 588-589, (1958) · Zbl 0081.02202
[5] Kostant, B., The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group, Amer. J. math., 81, 973-1032, (1959) · Zbl 0099.25603
[6] Lusztig, G., Singularities, character formulas, and a q-analog of weight multiplicities, Astérisque, 101-102, 208-229, (1983) · Zbl 0561.22013
[7] Sigler, L.E., Fibonacciʼs liber abaci, (2002), Springer-Verlag New York
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.