Diamond module for the Lie algebra $\mathfrak{so}(2n+1,\mathbb C)$

Boujemaa Agrebaoui (1) , Didier Arnal (2) , Abdelkader Ben Hassine (3)

1 FSS - Faculté des Sciences de Sfax
2 IMB - Institut de Mathématiques de Bourgogne [Dijon]
3 Département de Mathematiques [Sfax]

The diamond cone is a combinatorial description for a basis of an indecomposable module for the nilpotent factor $\mathfrak n$ of a semi simple Lie algebra. After N. J. Wildberger who introduced this notion, this description was achevied for $\mathfrak{sl}(n)$, the rank $2$ semi-simple Lie algebras and $\mathfrak{sp}(2n)$. In the present work, we generalize these constructions to the Lie algebras $\mathfrak{so}(2n+1)$. The orthogonal semistandard Young tableaux were defined by M. Kashiwara and T. Nakashima, they form a basis for the shape algebra of $\mathfrak{so}(2n+1)$. Defining the notion of orthogonal quasistandard Young tableaux, we prove these tableaux give a basis for the diamond module for $\mathfrak{so}(2n+1)$.

Mots clés: Shape algebra; Semistandard Young tableaux; Quasistandard Young tableaux; Jeu de taquin

Domaines: Algèbres quantiques [math.QA]

Boujemaa Agrebaoui, Didier Arnal, Abdelkader Ben Hassine. Diamond module for the Lie algebra $\mathfrak{so}(2n+1,\mathbb C)$. 2012. ⟨hal-00723844⟩