Diamond representations of $\mathfrak{sl}(n)$

Abstract

In \cite{W}, there is a graphic description of any irreducible, finite dimensional $\mathfrak{sl}(3)$ module. This construction, called diamond representation is very simple and can be easily extended to the space of irreducible finite dimensional ${\mathcal U}\_q(\mathfrak{sl}(3))$-modules. In the present work, we generalize this construction to $\mathfrak{sl}(n)$. We show this is in fact a description of the reduced shape algebra, a quotient of the shape algebra of $\mathfrak{sl}(n)$. The basis used in \cite{W} is thus naturally parametrized with the so called quasi standard Young tableaux. To compute the matrix coefficients of the representation in this basis, it is possible to use Groebner basis for the ideal of reduced Pl\"{u}cker relations defining the reduced shape algebra.