no. 17-18
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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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. http://www.numdam.org/articles/10.1016/j.crma.2011.08.017/

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