Diamond representations of $\mathfrak{sl}(n)$

Didier Arnal; Nadia Bel Baraka; Norman J. Wildberger

doi:10.5802/ambp.222

Diamond representations of

𝔰𝔩 (n)

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

Annales mathématiques Blaise Pascal, Tome 13 (2006) no. 2, pp. 381-429.

Résumé

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

In the present work, we generalize this construction to $𝔰𝔩 (n)$ . We show it is in fact a description of the reduced shape algebra, a quotient of the shape algebra of $𝔰𝔩 (n)$ . 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.

MR Zbl

DOI : 10.5802/ambp.222

Affiliations des auteurs :

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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     title = {Diamond representations of $\mathfrak{sl}(n)$},
     journal = {Annales math\'ematiques Blaise Pascal},
     pages = {381--429},
     publisher = {Annales math\'ematiques Blaise Pascal},
     volume = {13},
     number = {2},
     year = {2006},
     doi = {10.5802/ambp.222},
     mrnumber = {2275452},
     zbl = {1120.17005},
     language = {en},
     url = {https://ambp.centre-mersenne.org/articles/10.5802/ambp.222/}
}

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Didier Arnal; Nadia Bel Baraka; Norman J. Wildberger. Diamond representations of $\mathfrak{sl}(n)$. Annales mathématiques Blaise Pascal, Tome 13 (2006) no. 2, pp. 381-429. doi : 10.5802/ambp.222. https://ambp.centre-mersenne.org/articles/10.5802/ambp.222/

Bibliographie
Cité par

[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 $𝔰𝔩 (3)$ (2005) (Submitted)

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