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PLÜCKER RELATIONS AND SPHERICAL VARIETIES: APPLICATION TO MODEL VARIETIES

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A general framework for the reduction of the equations defining classes of spherical varieties to (possibly infinite-dimensional) grassmannians is proposed. This is applied to model varieties of types A, B and C; in particular, a standard monomial theory for these varieties is presented.

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References

  1. И. H. Бepнштeйн, И. M. Гeльфанд, C. И. Гeльфанд, Moдeлu пpeдcmaвлeнuй гpуnn Лu, Тpyды сeм. И. Г. Пeтpoвcкoгo, вып. 2 (1976), 3–21. Engl. transl.: I. N. Bernstein, I. M. Gelfand, S. I. Gelfand, S.I. Models of representations of Lie groups, Sel. Math. Sov. 1 (1981), 121–142.

  2. N. Bourbaki, Groupes et Algèbres de Lie, Chap. 4–6, Hermann, Paris, 1968; Masson, Paris, 1981. Russian transl.: H. Бypбaки, Гpynnы u aлгeбpы Лu, Mиp, M., 1972.

  3. P. Bravi, S. Cupit-Foutou, Classification of strict wonderful varieties, Ann. Inst. Fourier (Grenoble) 60(2) (2010), 641–681.

    Article  MATH  MathSciNet  Google Scholar 

  4. P. Bravi, J. Gandini, A. Maffei, Projective normality of model varieties and related results, arXiv:1304.6352.

  5. R. Chirivì, LS algebras and Schubert varieties, Publications of the Scuola Normale Superiore, Pisa, 2007.

    Google Scholar 

  6. R. Chirivì, LS algebras and application to Schubert varieties, Transform. Groups 5 (2000), no. 3, 245–264.

    Article  MATH  MathSciNet  Google Scholar 

  7. R. Chirivì, P. Littelmann, A. Maffei, Equations defining symmetric varieties and affine Grassmannians, Int. Math. Res. Not. 2 (2009), 291–347.

    Google Scholar 

  8. R. Chirivì, A. Maffei, Projective normality of symmetric varieties, Duke Math. Journal 122 (2004), no. 1, 93–123.

    Article  MATH  Google Scholar 

  9. C. DeConcini, D. Eisenbud, C. Procesi, Young diagrams and determinantal varieties, Invent. math. 56 (1980), 129–165.

    Article  MATH  MathSciNet  Google Scholar 

  10. D. Eisenbud, Commutative Algebra. With a View Toward Algebraic Geometry, Graduate Texts in Mathematics, Vol. 150, Springer-Verlag, New York, 1995.

  11. И. M. Гeльфaнд, A. B. Зeлeвинcкий, Moдeлu npeдcmaвлeнuй клaccuчecкux гpynn u ux cкpыmыe cuммempuu, Фyнкц. aнaлиз и eгo пpил. 18 (1984), no. 3, 14–31. Engl. transl.: I. M. Gel’fand, A. V. Zelevinskiĭ, Models of representations of classical groups and their hidden symmetries, Funct. Analysis Appl. 18 (1984), no. 3, 183–198.

  12. I. M. Gel’fand, A. V. Zelevinsky, Representation models for classical groups and their higher symmetries, in: The Mathematical Heritage of Élie Cartan (Lyon, 1984), Astérisque 1985, Numero Hors Serie, pp. 117–128.

  13. W. V. D. Hodge, Some enumerative results in the theory of forms, Proc. Cambridge Philos. Soc. (39) (1943), 22–30.

    Article  MATH  MathSciNet  Google Scholar 

  14. S. Kumar, Kac–Moody Groups, their Flag Varieties and Representation Theory, Progress in Mathematics, Vol. 204, Birkhäuser Boston, Boston, MA, 2002.

  15. V. Lakshmibai, C. Musili, C.S. Seshadri, Geometry of G/P, Bull. of the AMS 1 (1979), no. 2, 432–435.

    Article  MATH  MathSciNet  Google Scholar 

  16. P. Littelmann, Contracting modules and standard monomial theory for symmetrizable Kac-Moody algebras, J. Amer. Math. Soc. 11 (1998), no. 3, 551–567.

    Article  MATH  MathSciNet  Google Scholar 

  17. D. Luna, La variété magnifique modèle, J. Algebra 313 (2007), no. 1, 292–319.

    Article  MATH  MathSciNet  Google Scholar 

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Correspondence to ROCCO CHIRIVÌ.

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CHIRIVÌ, R., MAFFEI, A. PLÜCKER RELATIONS AND SPHERICAL VARIETIES: APPLICATION TO MODEL VARIETIES. Transformation Groups 19, 979–997 (2014). https://doi.org/10.1007/s00031-014-9285-6

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  • DOI: https://doi.org/10.1007/s00031-014-9285-6

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