## A geometric approach to Standard Monomial Theory

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- by M. Brion and V. Lakshmibai
- Represent. Theory
**7**(2003), 651-680 - DOI: https://doi.org/10.1090/S1088-4165-03-00211-5
- Published electronically: November 24, 2003
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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.## References

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## Bibliographic Information

**M. Brion**- Affiliation: Institut Fourier, UMR 5582 du CNRS, F-38402 Saint-Martin d’Hères Cedex
- MR Author ID: 41725
- Email: Michel.Brion@ujf-grenoble.fr
**V. Lakshmibai**- Affiliation: Department of Mathematics, Northeastern University, Boston, Massachusetts 02115-5096
- Email: lakshmibai@neu.edu
- Received by editor(s): November 8, 2001
- Received by editor(s) in revised form: September 12, 2003
- Published electronically: November 24, 2003
- © Copyright 2003 American Mathematical Society
- Journal: Represent. Theory
**7**(2003), 651-680 - MSC (2000): Primary 14M15, 20G05, 14L30, 14L40
- DOI: https://doi.org/10.1090/S1088-4165-03-00211-5
- MathSciNet review: 2017071