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The Development of Standard Monomial Theory-I

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A Tribute to C. S. Seshadri

Abstract

In this exposition, we will give a comprehensive report on some of the important highlights in the early phase of “standard monomial theory” (SMT) and its first applications as developed by Seshadri (with Musili, Lakshmibai and Littelmann) (built on the ideas of Hodge’s enumerative geometry, [22]). The later developments are given in [32] (appearing in this volume).

With gratseful homage to C.S. Seshadri, on his 70th birthday.

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References

  1. S. Abeasis, On the Pliicker relations for the Grassmann Varieties, Adv. in Math., 36(1980), 277–282.

    Article  MathSciNet  MATH  Google Scholar 

  2. S. Abeasis and A. Del Era, Degenerations for the Representations of a Quiver of Type Am, J. of Alg., 93(1985), 376–412.

    Article  MathSciNet  MATH  Google Scholar 

  3. S.S. Abhyankar, Enumerative Combinatorics of Young Tableaux, Marcel- Dekker, NY (1988).

    MATH  Google Scholar 

  4. A. Borel, Linear Algebraic Groups, (second enlarged edition), Graduate Texts in Mathematics, Springer (1991).

    Book  MATH  Google Scholar 

  5. A. Borel, Linear Representations of Semi-simple Algebraic Groups, AMS Proc. Symp. Pure Math., Algebraic Geometry (Areata), 29(1975), 421–440.

    Article  MathSciNet  MATH  Google Scholar 

  6. N. Bourbaki, Algebre Commutative, Elements de Math., 27,28,30,31, Hermann, Paris (1964–69); English trans: Addison Wesley, London (1972).

    Google Scholar 

  7. N. Bourbaki, Groupes et Algebres de Lie, Chapitres 4, 5 et 6, Hermann, Paris (1968).

    MATH  Google Scholar 

  8. J. Carrell, On the Smooth Points of a Schubert Variety, CMS Conference Proceedings, 16(1994), 15–24; Proceedings of the Conference on “Representations of Groups: Lie, Algebraic, Finite and Quantum“, Banff, Alberta 1994).

    MathSciNet  MATH  Google Scholar 

  9. C. Chevalley, Sur les Decompositions Cellulaires des Espaces G/B (with a oreword by A. Borel), Proc. Symp. Pure Math., (Part 1, Algebraic Groups and their Generalizations: Classical Methods (Univ. Park, PA: 1991)), 56(1994), 1–23.

    MATH  Google Scholar 

  10. M.S. Datt, The Minuscule Grassmannian, preprint.

    Google Scholar 

  11. C. DeConcini, D. Eisenbud and C. Procesi, Young Diagrams and Determi-nantal Varieties, Invent, math. 56(1980), 129–165.

    Article  MathSciNet  MATH  Google Scholar 

  12. C. DeConcini, D. Eisenbud and C. Procesi, Hodge Algebras, Asterisque 91(1982).

    Google Scholar 

  13. C. DeConcini and C. Procesi, A Characteristic free approach to Invariant Theory, Adv. in Math., 21(1976), 330–354.

    Article  MathSciNet  MATH  Google Scholar 

  14. C. DeConcini and E. Strickland, On the Variety of Complexes, Adv. in Math., 41(1981), 57–77.

    Article  MathSciNet  Google Scholar 

  15. M. Demazure, Desingularisation des Varietes de Schubert Generalises, Ann. Sci. Ec. Norm. Sup., t.7(1974), 53–88.

    Article  MATH  Google Scholar 

  16. V.V. Deodhar, On Some Geometric aspects of Bruhat Orderings, I. A finer decomposition of Bruhat cells. Invent, math. 79(1985), 499–511.

    Article  MATH  Google Scholar 

  17. D. Eisenbud, Introduction to Algebras with Straightening Laws (in Ring Theory and Algebra - III), Proc. of the Third Oklohoma Conf, Ed. B.R. McDonald, Marcel Dekker, NY (1980).

    Google Scholar 

  18. P. Gabriel and V. Dlab (Eds.), Proc. ICRA-II, Springer LNM 831, Ottawa (1979).

    Google Scholar 

  19. N. Gonciulea and V. Lakshmibai, Singular Loci of Ladder Determinantal Varieties and Schubert Varieties, J. Alg., 229(2000), 463–497.

    Article  MathSciNet  MATH  Google Scholar 

  20. R. Hartshorne, Appendix to Ample Vector Bundles, Publ. Math. IHES, 29(1966), 63–94.

    MATH  Google Scholar 

  21. M. Hochster and J.A. Eagon, Cohen-Macaulay Rings, Invariant Theory, and the Generic Perfection of Determinantal Loci, Amer. J. Math., 93(1971), 1020–1058.

    Article  MathSciNet  MATH  Google Scholar 

  22. W.V.D. Hodge, Some Enumerative results in the Theory of Forms, Proc. Camb. Phil. Soc, 39(1943), 22–30.

    Article  MathSciNet  MATH  Google Scholar 

  23. W.V.D. Hodge and D. Pedoe, Methods of Algebraic Geometry, Vol. II, Camb. Univ. Press, Cambridge (1952).

    MATH  Google Scholar 

  24. J.I. Igusa, On the Arithmetic Normality of the Grassmann Variety, Proc. Nat. Acad. Sci., USA, 40(1954), 309–313.

    Article  MathSciNet  MATH  Google Scholar 

  25. G. Kempf and A. Ramanathan, Multi-cones over Schubert Varieties, Invent. math., 87(1987), 353–363.

    Article  MathSciNet  MATH  Google Scholar 

  26. S.L. Kleiman, Problem 15: Rigourus Foundations of Schubert’s Enumerative Calculus, in Mathematical Developments arising from Hilbert Problems, AMS Proc. Symp. Pure Math. 28(1976), 445–482.

    Article  Google Scholar 

  27. B. Kostant, Lie Algebra Cohomology and the Generalised Borel-Weil Theorem, Ann. Math., 74(1961), 329–387.

    Article  MathSciNet  MATH  Google Scholar 

  28. V. Lakshmibai, Singular loci of Schubert Varieties for classical groups, Bull. Amer. Math. Soc, 16(1987), 83–90.

    Article  MathSciNet  MATH  Google Scholar 

  29. V. Lakshmibai, On Tangent Spaces to Schubert Varieties - I, J. Alg., 224(2000), 167–197.

    Article  MathSciNet  MATH  Google Scholar 

  30. V. Lakshmibai, On Tangent Spaces to Schubert Varieties - II, J. Alg., 230(2000), 222–244.

    Article  MathSciNet  MATH  Google Scholar 

  31. V. Lakshmibai, Singular Loci of Varieties of Complexes, J. Pure and Appl. Alg., 152(2000), 217–230.

    Article  MathSciNet  MATH  Google Scholar 

  32. V. Lakshmibai, The Development of Standard Monomial Theory - II, (this volume).

    Google Scholar 

  33. V. Lakshmibai, C. Musili and C.S. Seshadri, Geometry of G/P-III, Proc. Ind. Acad. Sci. (Math. Sci.), 87A(1978), 93–177.

    MATH  Google Scholar 

  34. V. Lakshmibai, C. Musili and C.S. Seshadri, Geometry of G/P - IV, Proc. Ind. Acad. Sci. (Math. Sci.), 88A(1979), 279–362.

    Article  MATH  Google Scholar 

  35. V. Lakshmibai, C. Musili and C.S. Seshadri, Geometry of G/P, Bulletin (New Series) of the AMS, 1(1979), 432–435.

    Article  MATH  Google Scholar 

  36. V. Lakshmibai and B. Sandhya, Criterion for Smoothness of Schubert Varieties in SL(n)/B, Proc. Ind. Acad. Sci. (Math. Sci.), 100(1990), 45–52.

    Article  MathSciNet  MATH  Google Scholar 

  37. V. Lakshmibai and C.S. Seshadri, Geometry of G/P — II, Proc. Ind. Acad. Sci. (Math. Sci.), 87A(1978), 1–54.

    Article  MATH  Google Scholar 

  38. V. Lakshmibai and C.S. Seshadri, Singular locus of a Schubert variety, Bull. Amer. Math. Soc, 2(1984), 363–366.

    Article  MathSciNet  MATH  Google Scholar 

  39. V. Lakshmibai and C.S. Seshadri, Geometry of G/P - V, J. of Alg., 100(1986), 462–557.

    Article  MATH  Google Scholar 

  40. V. Lakshmibai and C.S. Seshadri, Standard Monomial Theory, Proc Hyderabad Conference on Algebraic Groups, (Eds., S. Ramanan et al), Manoj Prakashan, Madras (1991), 279–323.

    Google Scholar 

  41. V. Lakshmibai and J. Weyman, Multiplicities of points on a Schubert Variety in a Minuscule G/P, Adv. Math., 84(1990), 179–208.

    Article  MathSciNet  MATH  Google Scholar 

  42. V. Mehta and A. Ramanathan, Probenius Splitting and Cohomology Vanishing for Schubert Varieties, Ann. Math., 122(1985), 27–40.

    Article  MathSciNet  MATH  Google Scholar 

  43. V.B. Mehta and A. Ramanathan, Schubert Varieties in G/B x G/B, Comp. Math. 67(1988), 355–358.

    MathSciNet  MATH  Google Scholar 

  44. S.B. Mulay, Determinantal Loci and the Flag Variety, Adv. in Math., 74(1989), 1–30.

    Article  MathSciNet  MATH  Google Scholar 

  45. C. Musili, Postulation Formula for Schubert Varieties, J. Ind. Math. Soc, 36(1972), 143–171.

    MathSciNet  MATH  Google Scholar 

  46. C. Musili, Some Properties of Schubert Varieties, J. Ind. Math. Soc, 38(1974), 131–145.

    MathSciNet  MATH  Google Scholar 

  47. C. Musili, A note on the Variety of Projectors, J. Pure and Applied Algebra, 74(1991), 73–84.

    Article  MathSciNet  MATH  Google Scholar 

  48. C. Musili, Applications of Standard Monomial Theory, Proc. Hyderabad Conference on Algebraic Groups, (Eds., S. Ramanan et al), Manoj Prakashan, Madras (1991), 381–406.

    Google Scholar 

  49. C. Musili, Algebraic Geometry for Beginners, TRIM-20, Hindustan Book Agency (India) Ltd., New Delhi (2001).

    Book  MATH  Google Scholar 

  50. C. Musili and C.S. Seshadri, Standard Monomial Theory, in Séminaire d’Algébre Paul Dubreil et Marie-Paule Malliavin, Springer LNM, 867(1981), 441–478.

    Google Scholar 

  51. C. Musili and C.S. Seshadri, Schubert Varieties and the Variety of Complexes, Arithmetic and Geometry, Vol.11, (Papers dedicated to I.R. Shafare-vich on his 60th Birthday), Progress in Mathematics, Birkhauser, Basel, 36(1983), 329–359,

    Article  MathSciNet  MATH  Google Scholar 

  52. S. Ramanan and A. Ramanathan, Projective Normality of Flag Varieties and Schubert Varieties, Invent, math, 79(1985), 217–224.

    Article  MathSciNet  MATH  Google Scholar 

  53. A. Ramanathan, Equations defining Schubert Varieties and Frobenius Splitting of Diagonals, Publ. Math. IHES, 65(1987), 61–90.

    Article  MathSciNet  MATH  Google Scholar 

  54. C.S. Seshadri, Geometry of G/P - I, C.P. Ramanujam: A Tribute, 207 (Springer-Verlag), Published by TIFR (1978), 207–239.

    Google Scholar 

  55. C.S. Seshadri, Introduction to the Theory of Standard Monomials, Brandeis Lecture Notes 4, Waltham (1985).

    MATH  Google Scholar 

  56. H. Weyl, The Classical Groups, Princeton Univ. Press, Princeton, NJ (1946).

    MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Math. and Stat., University of Hyderabad, P.O. Central University, Hyderabad, 500 046, India

    C. Musili

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  1. C. Musili
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V. Lakshmibai V. Balaji V. B. Mehta K. R. Nagarajan K. Pranjape P. Sankaran R. Sridharan

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© 2003 Hindustan Book Agency

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Musili, C. (2003). The Development of Standard Monomial Theory-I. In: Lakshmibai, V., et al. A Tribute to C. S. Seshadri. Hindustan Book Agency, Gurgaon. https://doi.org/10.1007/978-93-86279-11-8_24

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