Article
Full entry |
PDF
(0.3 MB)
Feedback
PDF
(0.3 MB)
Feedback
Keywords:
rational surface; minuscule representation; polytope
rational surface; minuscule representation; polytope
Summary:
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.
References:
[1] Bourbaki, N.: Elements of Mathematics. Lie Groups and Lie algebras. Chapters 4-6. Springer, Berlin (2002). DOI 10.1007/978-3-540-89394-3 | MR 1890629 | Zbl 0983.17001
[2] Coxeter, H. S. M.: Regular and semiregular polytopes. II. Math. Z. 188 (1985), 559-591. DOI 10.1007/BF01161657 | MR 0774558 | Zbl 0547.52005
[3] Coxeter, H. S. M.: Regular and semi-regular polytopes. III. Math. Z. 200 (1988), 3-45. DOI 10.1007/BF01161745 | MR 0972395 | Zbl 0633.52006
[4] Coxeter, H. S. M.: The evolution of Coxeter-Dynkin diagrams. Nieuw Arch. Wiskd., IV. Ser. 9 (1991), 233-248. MR 1166143 | Zbl 0759.20013
[5] Demazure, M., Pinkham, H., (eds.), B. Teissier: Séminaire sur les Singularités des Surfaces. Centre de Mathématiques de l'Ecole Polytechnique, Palaiseau 1976-1977, Lecture Notes in Mathematics 777, Springer, Berlin (1980), French. DOI 10.1007/BFb0085872 | MR 0579026 | Zbl 0415.00010
[6] Donagi, R. Y.: Principal bundles on elliptic fibrations. Asian J. Math. 1 (1997), 214-223. DOI 10.4310/AJM.1997.v1.n2.a1 | MR 1491982 | Zbl 0927.14006
[7] Donagi, R.: Taniguchi lectures on principal bundles on elliptic fibrations. Integrable Systems and Algebraic Geometry Conf. Proc., Kobe/Kyoto, 1997, World Scientific, Singapore (1998), 33-46. MR 1672104 | Zbl 0963.14004
[8] Friedman, R., Morgan, J., Witten, E.: Vector bundles and $F$ theory. Commun. Math. Phys. 187 (1997), 679-743. DOI 10.1007/s002200050154 | MR 1468319 | Zbl 0919.14010
[9] Henry-Labordère, P., Julia, B., Paulot, L.: Borcherds symmetries in M-theory. J. High Energy Phys. 2002 (2002), No. 49, 31 pages. DOI 10.1088/1126-6708/2002/04/049 | MR 1911396
[10] Henry-Labordère, P., Julia, B., Paulot, L.: Real Borcherds superalgebras and M-theory. J. High Energy Phys. 2003 (2003), No. 60, 21 pages. DOI 10.1088/1126-6708/2003/04/060 | MR 1989546
[11] Lee, J.-H.: Gosset polytopes in Picard groups of del Pezzo surfaces. Can. J. Math. 64 (2012), 123-150. DOI 10.4153/CJM-2011-063-6 | MR 2932172 | Zbl 1268.14038
[12] Lee, J.-H.: Contractions of del Pezzo surfaces to $\mathbb P^{2}$ or $\mathbb P^{1}\times\mathbb P^{1}$. Rocky Mt. J. Math. 46 (2016), 1263-1273. DOI 10.1216/RMJ-2016-46-4-1263 | MR 3563181 | Zbl 06642645
[13] Leung, N. C.: ADE-bundle over rational surfaces, configuration of lines and rulings. Available at arXiv:math.AG/0009192.
[14] Leung, N. C., Xu, M., Zhang, J.: Kac-Moody $\widetilde E_k$-bundles over elliptic curves and del Pezzo surfaces with singularities of type $A$. Math. Ann. 352 (2012), 805-828. DOI 10.1007/s00208-011-0661-4 | MR 2892453 | Zbl 1242.14036
[15] Leung, N. C., Zhang, J.: Moduli of bundles over rational surfaces and elliptic curves. I: Simply laced cases. J. Lond. Math. Soc., II. Ser. 80 (2009), 750-770. DOI 10.1112/jlms/jdp053 | MR 2559127 | Zbl 1188.14025
[16] Looijenga, E.: Root systems and elliptic curves. Invent. Math. 38 (1976), 17-32. DOI 10.1007/BF01390167 | MR 0466134 | Zbl 0358.17016
[17] Manin, Y. I.: Cubic Forms: Algebra, Geometry, Arithmetic. North-Holland Mathematical Library 4, North-Holland Publishing Company, Amsterdam; American Elsevier Publishing Company, New York (1974). MR 0460349 | Zbl 0277.14014
[18] Manivel, L.: Configurations of lines and models of Lie algebras. J. Algebra 304 (2006), 457-486. DOI 10.1016/j.jalgebra.2006.04.029 | MR 2256401 | Zbl 1167.17001
[19] Seshadri, C. S.: 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), 207-239. MR 0541023 | Zbl 0447.14010
Search
Partner of
