# Diamond representations of rank two semisimple Lie algebras

Abstract : 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 semi-simple Lie algebra $\mathfrak g$, using the restrictions to $N$ of the simple finite dimensional modules of $\mathfrak g$. Such a description is given in \cite{[ABW]}, for the cas $\mathfrak g=\mathfrak{sl}(n)$. Here, we give the analog for the rank 2 semi simple 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 reduced shape algebra of the shape algebra for $\mathfrak g$. Basis for the shape algebra are known, for instance the so called semi standard Young tableaux (see \cite{[ADLMPPrW]}). We select among the semi standard tableaux, the so called quasi standard ones which define a kind basis for the reduced shape algebra.
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https://hal.archives-ouvertes.fr/hal-00300612
Contributor : Didier Arnal <>
Submitted on : Monday, July 21, 2008 - 2:56:55 PM
Last modification on : Sunday, June 2, 2019 - 5:48:02 PM
Long-term archiving on : Monday, May 31, 2010 - 8:42:13 PM

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### Identifiers

• HAL Id : hal-00300612, version 1
• ARXIV : 0807.3256

### Citation

Boujemaa Agrebaoui, Didier Arnal, Olfa Khlifi. Diamond representations of rank two semisimple Lie algebras. Journal of Lie Theory, 2009, 19 (2), pp.339-370. ⟨hal-00300612⟩

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