Diamond representations of rank two semisimple Lie algebras
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 dime...
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description | 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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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.</description><identifier>EISSN: 2331-8422</identifier><language>eng</language><publisher>Ithaca: Cornell University Library, arXiv.org</publisher><subject>Algebra ; Combinatorial analysis ; Diamonds ; Functions (mathematics) ; Lie groups ; Mathematical analysis ; Polynomials ; Representations</subject><ispartof>arXiv.org, 2008-07</ispartof><rights>2008. 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subjects | Algebra Combinatorial analysis Diamonds Functions (mathematics) Lie groups Mathematical analysis Polynomials Representations |
title | Diamond representations of rank two semisimple Lie algebras |
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