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Journal of Lie Theory 19 (2009), No. 2, 339--370
Copyright Heldermann Verlag 2009

Diamond Representations for Rank Two Semisimple Lie Algebras

Boujemaa Agrebaoui
Dept. of Mathematics, Faculty of Sciences at Sfax, Route Soukra, B.P. 1171, 3000 Sfax, Tunisia

Didier Arnal
Institut de Mathématique, Université de Bourgogne, U.F.R. Sciences et Techniques, B.P. 47870, 21078 Dijon, France

Olfa Khlifi
Dept. of Mathematics, Faculty of Sciences at Sfax, Route Soukra, B.P. 1171, 3000 Sfax, Tunisia


\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.

Keywords: Rank two semisimple Lie algebras, representations, Young tableaux.

MSC: 05E10, 05A15, 17B10

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