Abstract
We give a closed-form formula for the Hilbert function of the tangent cone at the identity of a Schubert variety X in the Grassmannian in both group theoretic and combinatorial terms. We also give a formula for the multiplicity of X at the identity, and a Gröbner basis for the ideal defining X(w) ∩ O − as a closed subvariety of O −, where O − is the opposite cell in the Grassmannian. We give conjectures for the Hilbert function and multiplicity at points other than the identity.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
S.S. Abhyankar, Enumerative combinatorics of Young tableaux, Monographs and Textbooks in Pure and Applied Mathematics, 115. Marcel Dekker, Inc., New York (1988).
N. Bourbaki, Groupes et Algèbres de Lie, Chapitres 4, 5 et 6, Hermann, Paris (1968).
D. Cox, J. Little and D. O’Shea,Ideals, varieties and algorithms, Springer-Verlag, New York, (1992).
D. Eisenbud, Commutative algebra with a view toward Algebraic Geometry, Springer-Verlag, GTM, 150.
N. Gonciulea and V. Lakshmibai, Flag varieties, to appear in Hermann-Actualites Mathematiques.
J. Herzog and N.V. Trung, Gröbner bases and multiplicity of determinantal and Pfaffian ideals, Adv. Math., 96 (1992), 1–37.
V. Lakshmibai and J. Weyman, Multiplicities of points on a Schubert variety in a minuscule G/P, Adv. in Math., 84 (1990), 179–208.
C. Musili, Postulation formula for Schubert varieties, J. Indian Math. Soc., 36 (1972), 143–171.
J. Rosenthal and Zelevinsky, An explicit formula for the multiplicity of points on a classical Schubert variety, preprint (1998).
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2004 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Kreiman, V., Lakshmibai, V. (2004). Multiplicities of Singular Points in Schubert Varieties of Grassmannians. In: Christensen, C., Sathaye, A., Sundaram, G., Bajaj, C. (eds) Algebra, Arithmetic and Geometry with Applications. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-18487-1_31
Download citation
DOI: https://doi.org/10.1007/978-3-642-18487-1_31
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-00475-2
Online ISBN: 978-3-642-18487-1
eBook Packages: Springer Book Archive