Combinatorics/Lie Algebras
On the adjoint representation of sln and the Fibonacci numbers
[Sur la représentation adjointe de sln et les nombres de Fibonacci]

Présenté par : Michèle Vergne

Harris, Pamela E.1
1 University of Wisconsin, Milwaukee Department of Mathematical Sciences, P.O. Box 0413, Milwaukee, WI 53201, USA
Comptes Rendus. Mathématique, Tome 349 (2011) no. 17-18, pp. 935-937.

Nous décomposons la représentation adjointe de slr+1=slr+1(C) par une approche purement combinatoire basée sur lʼintroduction dʼun certain sous-ensemble du groupe de Weyl appelé Weyl alternation set associé à une paire de poids intégraux dominants. La cardinalité de Weyl alternation set associé à la plus haute racine et au poids zéro de slr+1 est donnée par le nombre rth de Fibonacci. Nous obtenons alors les exposants de slr+1 de ce point de vue.

We decompose the adjoint representation of slr+1=slr+1(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 slr+1 is given by the rth Fibonacci number. We then obtain the exponents of slr+1 from this point of view.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2011.08.017
Harris, Pamela E. 1

1 University of Wisconsin, Milwaukee Department of Mathematical Sciences, P.O. Box 0413, Milwaukee, WI 53201, USA 
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     title = {On the adjoint representation of $ {\mathfrak{sl}}_{n}$ and the {Fibonacci} numbers},
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Harris, Pamela E. On the adjoint representation of $ {\mathfrak{sl}}_{n}$ and the Fibonacci numbers. Comptes Rendus. Mathématique, Tome 349 (2011) no. 17-18, pp. 935-937. doi : 10.1016/j.crma.2011.08.017. https://www.numdam.org/articles/10.1016/j.crma.2011.08.017/

[1] C. Cochet, Vector partition function and representation theory, Conference Proceedings Formal Power Series and Algebraic Combinatorics, Taormina, Sicile, 2005, 12 pages.

[2] Goodman, R.; Wallach, N.R. Symmetry, Representations and Invariants, Springer, New York, 2009

[3] Humphreys, J.E. Reflection Groups and Coxeter Groups, Cambridge University Press, Cambridge, 1990

[4] Kostant, B. A formula for the multiplicity of a weight, Proc. Natl. Acad. Sci. USA, Volume 44 (1958), pp. 588-589

[5] Kostant, B. The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group, Amer. J. Math., Volume 81 (1959), pp. 973-1032

[6] Lusztig, G. Singularities, character formulas, and a q-analog of weight multiplicities, Astérisque, Volume 101–102 (1983), pp. 208-229

[7] Sigler, L.E. Fibonacciʼs Liber Abaci, Springer-Verlag, New York, 2002

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