Czechoslovak Mathematical Journal

, Volume 67, Issue 2, pp 397–415

Polytopes, quasi-minuscule representations and rational surfaces



We describe the relation between quasi-minuscule representations, polytopes and Weyl group orbits in Picard lattices of rational surfaces. As an application, to each quasi-minuscule representation we attach a class of rational surfaces, and realize such a representation as an associated vector bundle of a principal bundle over these surfaces. Moreover, any quasi-minuscule representation can be defined by rational curves, or their disjoint unions in a rational surface, satisfying certain natural numerical conditions.


rational surface minuscule representation polytope 

MSC 2010

14J26 14N20 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    N. Bourbaki: Elements of Mathematics, Lie Groups and Lie algebras. Chapters 4–6. Springer, Berlin, 2002.CrossRefMATHGoogle Scholar
  2. [2]
    H. S. M. Coxeter: Regular and semiregular polytopes. II. Math. Z. 188 (1985), 559–591.MathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    H. S. M. Coxeter: Regular and semi-regular polytopes. III. Math. Z. 200 (1988), 3–45.MathSciNetCrossRefMATHGoogle Scholar
  4. [4]
    H. S. M. Coxeter: The evolution of Coxeter-Dynkin diagrams. Nieuw Arch. Wiskd., IV. Ser. 9 (1991), 233–248.MathSciNetMATHGoogle Scholar
  5. [5]
    M. Demazure, H. Pinkham, B. Teissier (eds.): Séminaire sur les Singularités des Surfaces. Centre de Mathématiques de l’Ecole Polytechnique, Palaiseau 1976–1977, Lecture Notes in Mathematics 777, Springer, Berlin, 1980. (In French.)Google Scholar
  6. [6]
    R. Y. Donagi: Principal bundles on elliptic fibrations. Asian J. Math. 1 (1997), 214–223.MathSciNetCrossRefMATHGoogle Scholar
  7. [7]
    R. Donagi: Taniguchi lectures on principal bundles on elliptic fibrations. Integrable Systems and Algebraic Geometry. Conf. Proc., Kobe/Kyoto, 1997, World Scientific, Singapore, 1998, pp. 33–46.Google Scholar
  8. [8]
    R. Friedman, J. Morgan, E. Witten: Vector bundles and F theory. Commun. Math. Phys. 187 (1997), 679–743.CrossRefMATHGoogle Scholar
  9. [9]
    P. Henry-Labordère, B. Julia, L. Paulot: Borcherds symmetries in M-theory. J. High Energy Phys. 2002 (2002), No. 49, 31 pages.Google Scholar
  10. [10]
    P. Henry-Labordère, B. Julia, L. Paulot: Real Borcherds superalgebras and M-theory. J. High Energy Phys. 2003 (2003), No. 60, 21 pages.Google Scholar
  11. [11]
    J.-H. Lee: Gosset polytopes in Picard groups of del Pezzo surfaces. Can. J. Math. 64 (2012), 123–150.MathSciNetCrossRefMATHGoogle Scholar
  12. [12]
    J.-H. Lee: Contractions of del Pezzo surfaces to P2 or P1 × P1. Rocky Mt. J. Math. 46 (2016), 1263–1273.CrossRefGoogle Scholar
  13. [13]
    N. C. Leung: ADE-bundle over rational surfaces, configuration of lines and rulings. Available at arXiv:math. AG/0009192.Google Scholar
  14. [14]
    N. C. Leung, M. Xu, J. Zhang: Kac-Moody \({\tilde E_k}\) -bundles over elliptic curves and del Pezzo surfaces with singularities of type A. Math. Ann. 352 (2012), 805–828.MathSciNetCrossRefMATHGoogle Scholar
  15. [15]
    N. C. Leung, J. Zhang: Moduli of bundles over rational surfaces and elliptic curves. I: Simply laced cases. J. Lond. Math. Soc., II. Ser. 80 (2009), 750–770.MathSciNetCrossRefMATHGoogle Scholar
  16. [16]
    E. Looijenga: Root systems and elliptic curves. Invent. Math. 38 (1976), 17–32.MathSciNetCrossRefMATHGoogle Scholar
  17. [17]
    Y. I. Manin: Cubic Forms: Algebra, Geometry, Arithmetic. North-Holland Mathematical Library 4, North-Holland Publishing Company, Amsterdam; American Elsevier Publishing Company, New York, 1974.Google Scholar
  18. [18]
    L. Manivel: Configurations of lines and models of Lie algebras. J. Algebra 304 (2006), 457–486.MathSciNetCrossRefMATHGoogle Scholar
  19. [19]
    C. S. Seshadri: Geometry of G/P. I: Theory of standard monomials for minuscule representations. C. P. Ramanujam—A Tribute. Tata Inst. Fundam. Res., Stud. Math. 8, Springer, Berlin, 1978, pp. 207–239.Google Scholar

Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2017

Authors and Affiliations

  1. 1.Department of MathematicsEwha Womans UniversitySeoulKorea
  2. 2.Department of MathematicsSouthwest Jiaotong UniversityChengdu, SichuanP.R. China
  3. 3.Department of MathematicsSichuan UniversityChengdu, SichuanP.R. China

Personalised recommendations