Le c\^one diamant symplectique

TL;DR

This paper generalizes the combinatorial description of a basis in modules for semi-simple Lie algebras to the symplectic case, introducing symplectic quasi-standard Young tableaux to describe the diamond cone for sp(2n).

Contribution

It extends the diamond cone construction to sp(2n) Lie algebras using symplectic quasi-standard Young tableaux, building on prior work for other Lie types.

Findings

Defined symplectic quasi-standard Young tableaux.

Described the diamond cone for sp(2n) using these tableaux.

Connected the construction to the shape algebra of sp(2n).

Abstract

The diamond cone is a combinatorial description for a basis in a indecomposable module for the nilpotent factor n+ of a semi simple Lie algebra. After N.J. Wildberger who introduced this notion for sl(3), this description was achevied by N. Bel Baraka, N.J. Wildberger and D. A. for sl(n) and by B. Agrebaoui and ourselves for the rank 2 semi-simple Lie algebras. In the present work, we generalize these constructions to the Lie algebras sp(2n). The symplectic semi-standard Young tableaux were defined by C. de Concini, they form a basis for the shape algebra of sp(2n). We introduce here the notion of symplectic quasi-standard Young tableaux, these tableaux describe the diamond cone for sp(2n).