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A geometric approach to Standard Monomial Theory
Authors:
M. Brion and V. Lakshmibai
Journal:
Represent. Theory 7 (2003), 651-680
MSC (2000):
Primary 14M15, 20G05, 14L30, 14L40
Posted:
November 24, 2003
MathSciNet review:
2017071
Full-text PDF Free Access
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Abstract: We obtain a geometric construction of a ``standard monomial basis'' for the homogeneous coordinate ring associated with any ample line bundle on any flag variety. This basis is compatible with Schubert varieties, opposite Schubert varieties, and unions of intersections of these varieties. Our approach relies on vanishing theorems and a degeneration of the diagonal; it also yields a standard monomial basis for the multi-homogeneous coordinate rings of flag varieties of classical type.
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- 2.
- M. BRION and P. POLO: Large Schubert varieties, Represent. Theory 4 (2000), 97-126. MR 2001j:14066
- 3.
- C. CHEVALLEY: Sur les décompositions cellulaires des espaces
 (with a foreword by A. Borel), Proc. Sympos. Pure Math. 56, Part 1, Algebraic Groups and their Generalizations: Classical Methods (University Park, PA; 1991), Amer. Math. Soc., Providence, RI (1994), 1-23. MR 95e:14041
- 4.
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- 5.
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 -equivariant  -theory of generalized flag varieties, J. Differential Geom. 32 (1990), 549-603. MR 92c:19006
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- 10.
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- V. MEHTA and W. VAN DER KALLEN: On a Grauert-Riemenschneider theorem for Frobenius split varieties in characteristic
 , Invent. Math. 108 (1992), 11-13. MR 93a:14017
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- S. RAMANAN and A. RAMANATHAN: Projective normality of flag varieties and Schubert varieties, Invent. Math. 79 (1985), 217-234. MR 86j:14051
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- A. RAMANATHAN: Equations defining Schubert varieties and Frobenius splitting of diagonals, Pub. Math. IHES 65 (1987), 61-90. MR 88k:14032
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Additional Information
M. Brion
Affiliation:
Institut Fourier, UMR 5582 du CNRS, F-38402 Saint-Martin d’Hères Cedex
Email:
Michel.Brion@ujf-grenoble.fr
V. Lakshmibai
Affiliation:
Department of Mathematics, Northeastern University, Boston, Massachusetts 02115-5096
Email:
lakshmibai@neu.edu
DOI:
http://dx.doi.org/10.1090/S1088-4165-03-00211-5
PII:
S 1088-4165(03)00211-5
Received by editor(s):
November 8, 2001
Received by editor(s) in revised form:
September 12, 2003
Posted:
November 24, 2003
Article copyright:
© Copyright 2003 American Mathematical Society
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