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RESEARCH PRODUCT

Le cône diamant

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The diamond cone was introduced by N. J. Wildberger for the Lie algebra sl(n;R). It is a combinatoric presentation for the space C[N] of polynomials functions on the nilpotent factor N in the Iwasawa decomposition of SL(n;R). This presentation describes the natural layering of this indecomposable N-module. This basis can be indexeded by using some semi standard Young tableaux. We realize the algebra C[N] as a quotient of the shape algebra S_ for SL(n;R). Let us call reduced shape algebra this quotient. It is possible to select a family of semi standard Young tableaux, the quasi standard tableaux, in such a manner to get a basis for the reduced shape algebra. In the present thesis, this construction is extended to the case of rank 2 semi simple Lie algebras, then to the cas of the Lie algebras sp(2n), finally, to the case of the super simple Lie algebra sl(m; 1). In each case, we define the quasi standard Young tableaux, and show they define a good basis for the reduced shape algebra, either directly, or using an adapted version of the jeu de taquin defined by Schützenberger.