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Hilbert functions of points on Schubert varieties in the symplectic Grassmannian
Authors:
Sudhir R. Ghorpade and K. N. Raghavan
Journal:
Trans. Amer. Math. Soc. 358 (2006), 5401-5423
MSC (2000):
Primary 14M15, 13F50, 13A30
Posted:
July 20, 2006
MathSciNet review:
2238920
Full-text PDF Free Access
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Abstract: We give an explicit combinatorial description of the multiplicity as well as the Hilbert function of the tangent cone at any point on a Schubert variety in the symplectic Grassmannian.
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- 1.
- S. S. Abhyankar, Enumerative combinatorics of Young tableaux, Marcel Dekker, New York, 1988. MR 0926272 (89e:05011)
- 2.
- M. Brion and P. Polo, Generic singularities of certain Schubert varieties, Math. Z. 231 (1999), 301-324. MR 1703350 (2000f:14078)
- 3.
- S. Billey and V. Lakshmibai, Singular Loci of Schubert Varieties, Progress in Math., Vol. 182, Birkhäuser, Boston, 2000. MR 1782635 (2001j:14065)
- 4.
- A. Conca, Gröbner bases of ideals of minors of a symmetric matrix, J. Algebra 166 (1994), 406-421. MR 1279266 (95g:13012)
- 5.
- A. Conca and J. Herzog, On the Hilbert function of determinantal rings and their canonical module, Proc. Amer. Math. Soc. 122 (1994), 677-681. MR 1213858 (95a:13016)
- 6.
- C. De Concini, Symplectic Standard Tableaux, Adv. Math. 34 (1979), 1-27. MR 0547837 (80m:14036)
- 7.
- D. Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry, Grad. Texts in Math., Vol. 150, Springer-Verlag, New York, 1995. MR 1322960 (97a:13001)
- 8.
- W. Fulton and P. Pragacz, Schubert Varieties and Degeneracy Loci, Lect. Notes in Math., Vol. 1689, Springer-Verlag, Berlin, Heidelberg, 1995. MR 1639468 (99m:14092)
- 9.
- S. R. Ghorpade, Abhyankar's work on Young tableaux and some recent developments, Algebraic Geometry and its Applications (West Lafayette, IN, 1990), pp. 235-265, Springer-Verlag, New York, 1994. MR 1272033 (95h:05148)
- 10.
- S. R. Ghorpade, Young bitableaux, lattice paths and Hilbert functions, J. Statist. Plann. Inference 54 (1996), 55-60. MR 1406731 (98c:05165)
- 11.
- S. R. Ghorpade and C. Krattenthaler, The Hilbert series of Pfaffian rings, Algebra, Arithmetic and Geometry with Applications (West Lafayette, IN, 2000), pp. 337-356, Springer, Berlin, 2004. MR 2037100 (2005c:13010)
- 12.
- J. Herzog and N. V. Trung, Gröbner bases and multiplicity of determinantal and pfaffian ideals, Adv. Math. 96 (1992), 1-37. MR 1185786 (94a:13012)
- 13.
- A. Knutson and E. Miller, Gröbner Geometry of Schubert Polynomials, Ann. Math. (2) 161 (2005), 1245-1318. MR 2180402
- 14.
- V. Kodiyalam and K. N. Raghavan, Hilbert functions of points on Schubert varieties in Grassmannians, J. Algebra 270 (2003), 28-54. MR 2015929 (2005d:14067)
- 15.
- C. Krattenthaler, On multiplicities of points on Schubert varieties in Grassmannians, Sem. Lothar. Combin. 45 (2000/01), Art.B45c, 11pp (electronic). MR 1817336 (2002c:14080)
- 16.
- C. Krattenthaler, On multiplicities of points on Schubert varieties in Grassmannians II, J. Algebraic Combin. 22 (2005), 273-288. MR 2181366
- 17.
- V. Kreiman, Monomial Bases and Applications for Richardson and Schubert Varieties in Ordinary and Affine Grassmannians, Thesis, Northeastern University, 2003.
- 18.
- V. Kreiman and V. Lakshmibai, Multiplicities of Singular Points of Schubert Varieties of Grassmannians, Algebra, Arithmetic and Geometry with Applications (West Lafayette, IN, 2000), pp. 553-563, Springer, Berlin, 2004. MR 2037109 (2005c:14060)
- 19.
- V. Kreiman and V. Lakshmibai, Richardson Varieties in the Grassmannians, Contributions to Automorphic Forms, Geometry and Number Theory (Baltimore, MD, 2002), pp. 573-597, Johns Hopkins Univ. Press, Baltimore, MD, 2004. MR 2058620 (2005c:14059)
- 20.
- V. Lakshmibai, C. Musili and C. S. Seshadri, Geometry of
 - IV, Proc. Indian Acad. Sci. Sect. A Math. Sci. 88 (1979), 279-362. MR 0553746 (81g:14023d)
- 21.
- V. Lakshmibai and C. S. Seshadri, Geometry of
 - II, Proc. Indian Acad. Sci. Sect. A 87 (1978), 1-54. MR 0490244 (81g:14023b)
- 22.
- V. Lakshmibai and J. Weyman, Multiplicities of points on a Schubert variety in a minuscule
 , Adv. Math. 84 (1990), 179-208. MR 1080976 (92a:14058)
- 23.
- P. Littelmann, A Littlewood-Richardson rule for symmetrizable Kac-Moody algebras, Invent. Math. 116 (1994), 329-346. MR 1253196 (95f:17023)
- 24.
- P. Littelmann, Paths and root operators in representation theory, Ann. of Math. (2) 142 (1995), 499-525. MR 1356780 (96m:17011)
- 25.
- L. Manivel, Fonctions Symétriques, Polynômes de Schubert et Lieux de Dégénérescence, Cours Spécialisés No. 3, Soc. Math. de France, Paris, 1998. MR 1638048 (99k:05159)
- 26.
- J. Rosenthal and A. Zelevinsky, Multiplicities of points on Schubert varieties in Grassmannians, J. Algebraic Combin. 13 (2001), 213-218. MR 1826954 (2002d:14089)
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Additional Information
Sudhir R. Ghorpade
Affiliation:
Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India
Email:
srg@math.iitb.ac.in
K. N. Raghavan
Affiliation:
Institute of Mathematical Sciences, C.I.T. Campus, Taramani, Chennai 600113, India
Email:
knr@imsc.ernet.in
DOI:
http://dx.doi.org/10.1090/S0002-9947-06-04037-2
PII:
S 0002-9947(06)04037-2
Received by editor(s):
September 20, 2004
Posted:
July 20, 2006
Article copyright:
© Copyright 2006 American Mathematical Society
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