Diamond representations of 𝔰𝔩(n)
Annales mathématiques Blaise Pascal, Volume 13 (2006) no. 2, p. 381-429

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.

@article{AMBP_2006__13_2_381_0,
     author = {Arnal, Didier and Bel Baraka, Nadia and Wildberger, Norman J.},
     title = {Diamond representations of $\mathfrak{sl}(n)$},
     journal = {Annales math\'ematiques Blaise Pascal},
     publisher = {Annales math\'ematiques Blaise Pascal},
     volume = {13},
     number = {2},
     year = {2006},
     pages = {381-429},
     doi = {10.5802/ambp.222},
     mrnumber = {2275452},
     zbl = {1120.17005},
     language = {en},
     url = {http://www.numdam.org/item/AMBP_2006__13_2_381_0}
}
Arnal, Didier; Bel Baraka, Nadia; Wildberger, Norman J. Diamond representations of $\mathfrak{sl}(n)$. Annales mathématiques Blaise Pascal, Volume 13 (2006) no. 2, pp. 381-429. doi : 10.5802/ambp.222. http://www.numdam.org/item/AMBP_2006__13_2_381_0/

[1] Cox, D.; Little, J.; O’Shea, D. Ideals, varieties, and algorithms, Springer-Verlag, New York (1996) | Zbl 0861.13012

[2] Fulton, W.; Harris, J. Representation theory, Springer-Verlag, New York (1991) | MR 1153249 | Zbl 0744.22001

[3] Kashiwara, M. Bases cristallines des groupes quantiques, Soc. Math. France, Paris (2002) | MR 1997677 | Zbl 1066.17007

[4] Lancaster, G.; Towber, J. Representation-functors and flag-algebras for the classical groups, J. Algebra, Tome 59 (1979) | Article | MR 541667 | Zbl 0441.14013

[5] Varadarajan, V.S. Lie groups, Lie algebras, and their representations, Springer-Verlag, New York, Berlin (1984) | MR 746308 | Zbl 0955.22500

[6] Wildberger, N. Quarks, diamonds and representation of 𝔰𝔩(3) (2005) (Submitted)