arXiv:1208.3349v1 [math.QA] 16 Aug 2012 DIAMOND MODULE FOR THE LIE ALGEBRA so(2n +1, C) BOUJEMA ˆ A AGREBAOUI, DIDIER ARNAL AND ABDELKADER BEN HASSINE Abstract. The diamond cone is a combinatorial description for a basis of an inde- composable module for the nilpotent factor n of a semi simple Lie algebra. After N. J. Wildberger who introduced this notion, this description was achevied for sl(n), the rank 2 semi-simple Lie algebras and sp(2n). In the present work, we generalize these constructions to the Lie algebras so(2n + 1). The orthogonal semistandard Young tableaux were defined by M. Kashiwara and T. Nakashima, they form a basis for the shape algebra of so(2n + 1). Defining the notion of orthogonal quasistandard Young tableaux, we prove these tableaux give a basis for the diamond module for so(2n + 1). 1. Introduction Let g be a semisimple Lie algebra. The simple (finite dimensional) g-modules are characterized by their highest weight λ, each of them contains an unique (up to con- stant) vector v λ with weight λ, the g-action on v λ generates the corresponding simple module. The direct sum of all these modules is a natural algebra, the shape algebra of g. Consider now the nilpotent factor n in the Isawasa decomposition the Lie algebra g. It is natural to study nilpotent finite dimensional n-modules. They are generally indecomposable, if the module is generated by the action on an unique vector v, we say this module is monogenic. Each of the monogenic nilpotent module is a quotient of a well determined simple g-module (viewed as a n-module). The natural object corresponding to the shape algebra is now the diamond module, union of all these maximal monogenic modules. 2000 Mathematics Subject Classification. 20G05, 05A15, 17B10. Key words and phrases. Shape algebra, Semistandard Young tableaux, Quasistandard Young tableaux, Jeu de taquin. B. Agrebaoui, and A. Ben Hassine thank the Institut de Math´ ematiques de Bourgogne for his hospitality during his stays in Dijon, D. Arnal thanks the University of Sfax for its support and hospitality during his visits in Tunisia. 1