Diamond cone for $\mathfrak{sl}(m,n)$

Boujemaa Agrebaoui (1) , Didier Arnal (2) , Olfa Khlifi (1)

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

In this paper, we first study the shape algebra and the reduced shape algebra for the Lie superalgebra $\mathfrak{sl}(m,n)$. We define the quasistandard tableaux, their collection is the diamond cone for $\mathfrak{sl}(m,n)$, which is a combinatorial basis for the reduced shape algebra. We realize a bijection between the set of semistandard tableaux with shape $\lambda$ and the set of quasistandard tableaux with shape $\mu\leq\lambda$, by using the 'super jeu de taquin' on skew semistandard tableaux. This gives the compatibility of the diamond cone with the natural stratification of the reduced shape algebra.

Mots clés: Lie superalgebra; simples modules; Young tableaux

Domaines: Combinatoire [math.CO]; Théorie des représentations [math.RT]

Boujemaa Agrebaoui, Didier Arnal, Olfa Khlifi. Diamond cone for $\mathfrak{sl}(m,n)$. 2012. ⟨hal-00753100⟩