Diamond module for the Lie algebra \(\mathfrak{so}(2n+1,\mathbb C)\)

The diamond cone is a combinatorial description for a basis of an indecomposable module for the nilpotent factor \(\mathfrak n\) of a semi simple Lie algebra. After N. J. Wildberger who introduced this notion, this description was achevied for \(\mathfrak{sl}(n)\), the rank 2 semi-simple Lie algebra...

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Bibliographic Details
Published in:arXiv.org 2012-08
Main Authors: Agrebaoui, Boujemaa, Arnal, Didier, Abdelkader Ben Hassine
Format: Article
Language:English
Subjects:
Algebra
Combinatorial analysis
Diamonds
Lie groups
Modules
Quantum theory
Online Access:Get full text
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Summary:The diamond cone is a combinatorial description for a basis of an indecomposable module for the nilpotent factor \(\mathfrak n\) of a semi simple Lie algebra. After N. J. Wildberger who introduced this notion, this description was achevied for \(\mathfrak{sl}(n)\), the rank 2 semi-simple Lie algebras and \(\mathfrak{sp}(2n)\). In the present work, we generalize these constructions to the Lie algebras \(\mathfrak{so}(2n+1)\). The orthogonal semistandard Young tableaux were defined by M. Kashiwara and T. Nakashima, they form a basis for the shape algebra of \(\mathfrak{so}(2n+1)\). Defining the notion of orthogonal quasistandard Young tableaux, we prove these tableaux give a basis for the diamond module for \(\mathfrak{so}(2n+1)\).
ISSN:2331-8422