Journal of Lie Theory

no. 2
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, Volume 19 (2009) no. 2, pp. 339-370
\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.
DOI: 10.5802/jolt.556
Classification: 05E10, 05A15, 17B10
Keywords: Rank two semisimple Lie algebras, representations, Young tableaux

Boujemaa Agrebaoui  1 ; Didier Arnal  2 ; Olfa Khlifi  1

1 Dept. of Mathematics, Faculty of Sciences at Sfax, Route Soukra, B.P. 1171, 3000 Sfax, Tunisia
2 Institut de Mathématique, Université de Bourgogne, U.F.R. Sciences et Techniques, B.P. 47870, 21078 Dijon, France 
Boujemaa Agrebaoui; Didier Arnal; Olfa Khlifi. Diamond Representations for Rank Two Semisimple Lie Algebras. Journal of Lie Theory, Volume 19 (2009) no. 2, pp. 339-370. doi: 10.5802/jolt.556
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     title = {Diamond {Representations} for {Rank} {Two} {Semisimple} {Lie} {Algebras}},
     journal = {Journal of Lie Theory},
     pages = {339--370},
     year = {2009},
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     number = {2},
     doi = {10.5802/jolt.556},
     zbl = {1207.05216},
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