Diamond representations of $\mathrm{𝔰𝔩}\left(n\right)$
Annales mathématiques Blaise Pascal, Volume 13 (2006) no. 2, pp. 381-429.

In [6], there is a graphic description of any irreducible, finite dimensional $\mathrm{𝔰𝔩}\left(3\right)$ module. This construction, called diamond representation is very simple and can be easily extended to the space of irreducible finite dimensional ${𝒰}_{q}\left(\mathrm{𝔰𝔩}\left(3\right)\right)$-modules.

In the present work, we generalize this construction to $\mathrm{𝔰𝔩}\left(n\right)$. We show it is in fact a description of the reduced shape algebra, a quotient of the shape algebra of $\mathrm{𝔰𝔩}\left(n\right)$. The basis used in [6] is thus naturally parametrized with the so called quasi standard Young tableaux. To compute the matrix coefficients of the representation in this basis, it is possible to use Groebner basis for the ideal of reduced Plücker relations defining the reduced shape algebra.

DOI: 10.5802/ambp.222
Didier Arnal 1; Nadia Bel Baraka 1; Norman J. Wildberger 2

1 Institut de Mathématiques de Bourgogne UMR CNRS 5584 Université de Bourgogne U.F.R. Sciences et Techniques B.P. 47870 F-21078 Dijon Cedex France
2 School of Mathematics University of New South Wales Sydney 2052 Australia
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Didier Arnal; Nadia Bel Baraka; Norman J. Wildberger. Diamond representations of $\mathfrak{sl}(n)$. Annales mathématiques Blaise Pascal, Volume 13 (2006) no. 2, pp. 381-429. doi : 10.5802/ambp.222. https://ambp.centre-mersenne.org/articles/10.5802/ambp.222/

[1] D. Cox; J. Little; D. O’shea Ideals, varieties, and algorithms, Springer-Verlag, New York, 1996 | Zbl

[2] W. Fulton; J. Harris Representation theory, Springer-Verlag, New York, 1991 | MR | Zbl

[3] M. Kashiwara Bases cristallines des groupes quantiques, Soc. Math. France, Paris, 2002 | MR | Zbl

[4] G. Lancaster; J. Towber Representation-functors and flag-algebras for the classical groups, J. Algebra, Volume 59 (1979) | DOI | MR | Zbl

[5] V.S. Varadarajan Lie groups, Lie algebras, and their representations, Springer-Verlag, New York, Berlin, 1984 | MR | Zbl

[6] N. Wildberger Quarks, diamonds and representation of $\mathrm{𝔰𝔩}\left(3\right)$ (2005) (Submitted)

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