Diamond Representations for Rank Two Semisimple Lie Algebras

Boujemaa Agrebaoui¹ ; Didier Arnal² ; Olfa Khlifi¹

¹Dept. of Mathematics, Faculty of Sciences at Sfax, Route Soukra, B.P. 1171, 3000 Sfax, Tunisia
²Institut de Mathématique, Université de Bourgogne, U.F.R. Sciences et Techniques, B.P. 47870, 21078 Dijon, France

Journal of Lie Theory, Tome 19 (2009) no. 2, pp. 339-370

Résumé

\def\g{{\frak g}} The present work is a part of a larger program to construct explicit combinatorial models for the (indecomposable) regular representation of the nilpotent factor $N$ in the Iwasawa decomposition of a semisimple Lie algebra $\g$, using the restrictions to $N$ of the simple finite dimensional modules of $\g$. Such a description was given by D. Arnal, N. Bel Baraka and N.-J. Wildberger [{\it Diamond representations of $\frak{sl}(n)$}, International Journal of Algebra and Computation 13 (2006) 381--429] for the case $\g=\frak{sl}(n)$. Here, we perform the same construction for the rank 2 semisimple Lie algebras (of type $A_1 \times A_1$, $A_2$, $C_2$ and $G_2$). The algebra ${\mathbb C}[N]$ of polynomial functions on $N$ is a quotient, called the reduced shape algebra, of the shape algebra for $\g$. Bases for the shape algebra are known, for instance the so-called semistandard Young tableaux [see L.-W. Alverson, R.-G. Donnelly, S.-J. Lewis, M. McClard, R. Pervine, R.-A. Proctor, and N.-J. Wildberger, {\it Distributive lattice defined for representations of rank two semisimple Lie algebras}, ArXiv 0707.2421 v 1 (2007) 1--33] give an explicit basis. We select among the semistandard tableaux, the so-called quasistandard ones which define a kind basis for the reduced shape algebra.

Détail

Classification : 05E10, 05A15, 17B10
Mots-clés : Rank two semisimple Lie algebras, representations, Young tableaux