Journal of Lie Theory 30 (2020), No. 1, 277--303
Copyright Heldermann Verlag 2020
Jeu de Taquin and Diamond Cone for so(2n+1, C)
Boujemaâ Agrebaoui
Université de Sfax, Faculté des Sciences, Dép. de Mathématiques, 3000 Sfax, Tunisie
B.Agreba@fss.rnu.tn
Didier Arnal
Université de Bourgogne-Franche Comté, Institut de Mathématiques, U.F.R. Sciences et Techniques, 21078 Dijon, France
Didier.Arnal@u-bourgogne.fr
Abdelkader Ben Hassine
University of Bisha, Dept. of Mathematics, Faculty of Science and Arts, Belqarn, Sabt Al-Alaya 61985, Kingdom of Saudi Arabia
and: Université de Sfax, Faculté des Sciences, Dép. de Mathématiques, 3000 Sfax, Tunisie
benhassine.abdelkader@yahoo.fr
The diamond cone is a combinatorial description for a basis of a natural indecomposable n-module, where n is the nilpotent factor of a complex semisimple Lie algebra g. After N. J. Wildberger who introduced this notion, this description was achieved for g = sl(n), the rank 2 semisimple Lie algebras and g = sp(2n).
In this work, we generalize these constructions to the Lie algebra g = so(2n+1). The orthogonal semistandard Young tableaux were defined by M. Kashiwara and T. Nakashima, they index a basis for the shape algebra of so(2n+1). Defining the notion of orthogonal quasistandard Young tableaux, we prove that these tableaux describe a basis for a quotient of the shape algebra, the reduced shape algebra of so(2n+1).
Keywords: Shape algebra, semistandard Young tableau, quasistandard Young tableau, jeu de taquin.
MSC: 20G05, 05A15, 17B10.
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