Advertisement

Algebras and Representation Theory

, Volume 21, Issue 1, pp 219–237 | Cite as

Sliding Presentation of the Jeux de Taquin for Classical Lie Groups

  • Didier ArnalEmail author
  • Olfa Khlifi
Article
  • 60 Downloads

Abstract

The simple \(GL(n,\mathbb {C})\)-modules are described by using semistandard Young tableaux. Any semistandard skew tableau can be transformed into a well defined semistandard tableau by a combinatorial operation, the Schützenberger jeu de taquin. Associated to the classical Lie groups \(SP(2n,\mathbb {C})\), \(SO(2n+1,\mathbb {C})\), there are other notions of semistandard Young tableaux and jeux de taquin. In this paper, we study these various jeux de taquin, proving that each of them has a simple and explicit formulation as a step-by-step sliding. Any of these jeux de taquin is the restriction of the orthogonal one, associated to \(SO(2n+1,\mathbb {C})\).

Keywords

Jeu de taquin Semistandard young tableaux Classical lie groups 

Mathematics Subject Classification (2010)

17B20 05E10 05E15 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Agrebaoui, B., Arnal, D., Ben Hassine, A.: Diamond module for the Lie algebra \(\mathfrak {so}\)(2n+1, \(\mathbb {C}\)). arXiv:1208.3349v1 (2012)
  2. 2.
    Agrebaoui, B., Arnal, D., Ben Hassine, A.: Jeu de taquin and diamond cone for (super) Lie algebras. Bull. Sci. Math. 139(1), 75–113 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Arnal, D., Bel Baraka, N., Wildberger, N.: Diamond representations of \(\mathfrak {sl}(n)\). Ann. Math. Blaise Pascal 13(2), 381–429 (2006)Google Scholar
  4. 4.
    Arnal, D., Khlifi, O.: Le cône diamant symplectique. Bull. Sci. Math. 134 (6), 635–663 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bona, M.: Handbook of Enumerative Combinatorics, Discrete Mathematics and Its Applications, CRC Press. isbn:9781482220865 (2015)Google Scholar
  6. 6.
    Lecouvey, C.: Schensted-type correspondences, Plactic Monoid and Jeu de Taquin for Type c n. J. Algebra 247(2), 295–331 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Lecouvey, C.: Schensted-type correspondences and plactic monoids for types b n and d n. J. Algebraic Comb. 18(2), 99–133 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Sagan, B.E.: Shifted tableaux, Schur Q-functions, and a conjecture of R. Stanley. J. Combin. Theory Ser. A 45(1), 62–103 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Sheats, J.T.: A symplectic jeu de taquin bijection between the tableaux of King and of De Concini. Trans. Amer. Math. Soc. 351(9), 3569–3607 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Schützenberger, M.-P.: La correspondance de Robinson. In: Foata, D. (ed.) Combinatoire et représentation du groupe symétrique: Actes Table Ronde CNRS, University of Louis-Pasteur Strasbourg, Strasbourg, pp. 59–113 (1976). Springer, Lecture Notes in Math., no 579 (1977)Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  1. 1.Institut de Mathématiques de Bourgogne, UMR CNRS 5584Université de Bourgogne Franche ComtéDijon CedexFrance
  2. 2.Unité de recherche UR 09-06, Faculté des SciencesUniversité de SfaxSfaxTunisie

Personalised recommendations