Hilbert functions of points on Schubert varieties in orthogonal Grassmannians
K.N. Raghavan
and Shyamashree Upadhyay
Institute of Mathematical Sciences, C. I. T. Campus, Chennai 600113, India
DOI: 10.1007/s10801-009-0188-x
Abstract
Given a point on a Schubert variety in an orthogonal Grassmannian, we compute the multiplicity, more generally the Hilbert function. We first translate the problem from geometry to combinatorics by applying standard monomial theory. The solution of the resulting combinatorial problem forms the bulk of the paper. This approach has been followed earlier to solve the same problem for Grassmannians and symplectic Grassmannians.
As an application, we present an interpretation of the multiplicity as the number of non-intersecting lattice paths of a certain kind. A more important application, although it does not appear here but elsewhere, is to the computation of the initial ideal, with respect to certain convenient monomial orders, of the ideal of the tangent cone to the Schubert variety.
Pages: 355–409
Keywords: keywords orthogonal Grassmannian; Schubert variety; Hilbert function; multiplicity; Pfaffian ideal; O-domination; O-depth
Full Text: PDF
References
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2. De Negri, E.: Some results on Hilbert series and a-invariant of Pfaffian ideals. Math. J. Toyama Univ. 24, 93-106 (2001)
3. Ghorpade, S.R., Krattenthaler, C.: The Hilbert series of Pfaffian rings. In: Algebra, arithmetic and geometry with applications, pp. 337-356. West Lafayette, IN, 2000, Springer, Berlin (2004)
4. Ghorpade, S.R., Raghavan, K.N.: Hilbert functions of points on Schubert varieties in the Symplectic Grassmannian. Trans. Am. Math. Soc. 358, 5401-5423 (2006)
5. Herzog, J., Trung, N.V.: Gröbner bases and multiplicity of determinantal and Pfaffian ideals. Adv. Math. 96(1), 1-37 (1992)
6. Ikeda, T., Naruse, H.: Excited Young diagrams and equivariant Schubert calculus. Trans. Am. Math. Soc. Electronically published on April 30 (2009).
7. Kodiyalam, V., Raghavan, K.N.: Hilbert functions of points on Schubert varieties in Grassmannians. J. Algebra 270(1), 28-54 (2003)
8. Krattenthaler, C.: On multiplicities of points on Schubert varieties in Grassmannians. II. J. Algebraic Combin. 22(3), 273-288 (2005)
9. Kreiman, V.: Monomial bases and applications for Schubert and Richardson varieties in ordinary and affine Grassmannians. Ph.D. Thesis, Northeastern University (2003)
10. Kreiman, V.: Local properties of Richardson varieties in the Grassmannian via a bounded Robinson-Schensted-Knuth correspondence. J. Algebraic Combin. 27(3), 351-382 (2008)
11. Kreiman, V., Lakshmibai, V.: Multiplicities of singular points in Schubert varieties of Grassmannians. In: Algebra, arithmetic and geometry with applications, West Lafayette, IN, 2000, pp. 553-563. Springer, Berlin (2004)
12. Kreiman, V., Lakshmibai, V.: Richardson varieties in the Grassmannian. In: Contributions to automorphic forms, geometry, and number theory, pp. 573-597. Johns Hopkins Univ. Press, Baltimore (2004)
13. Lakshmibai, V., Seshadri, C.S.: Geometry of G/P . II. The work of de Concini and Procesi and the basic conjectures. Proc. Indian Acad. Sci. Sect. A 87(2), 1-54 (1978)
14. Lakshmibai, V., Seshadri, C.S.: Geometry of G/P . V. J. Algebra 100(2), 462-557 (1986)
15. Lakshmibai, V., Weyman, J.: Multiplicities of points on a Schubert variety in a minuscule G/P . Adv. Math. 84(2), 179-208 (1990)
16. Littelmann, P.: Contracting modules and standard monomial theory for symmetrizable Kac-Moody algebras. J. Amer. Math. Soc. 11(3), 551-567 (1998)
17. Raghavan, K.N., Sankaran, P.: A new approach to standard monomial theory for classical groups. Transform. Groups 3(1), 57-73 (1998)
18. Raghavan, K.N., Upadhyay, S.: Initial ideals of tangent cones to Schubert varieties in orthogonal Grassmannians. J. Combin. Theory Ser. A 116(3), 663-683 (2009).
19. Seshadri, C.S.: Geometry of G/P . I. Theory of standard monomials for minuscule representations. In: C.P. Ramanujam-a tribute. Tata Inst. Fund. Res. Studies in Math., vol. 8, pp. 207-239. Springer, Berlin (1978)
20. Upadhyay, S.: Schubert varieties in the Orthogonal Grassmannian. Ph.D. Thesis, Chennai Mathematical Institute (2008).
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