A geometric approach to Standard Monomial Theory
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- by M. Brion and V. Lakshmibai PDF
- Represent. Theory 7 (2003), 651-680 Request permission
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
- Michel Brion, Positivity in the Grothendieck group of complex flag varieties, J. Algebra 258 (2002), no. 1, 137–159. Special issue in celebration of Claudio Procesi’s 60th birthday. MR 1958901, DOI 10.1016/S0021-8693(02)00505-7
- Michel Brion and Patrick Polo, Large Schubert varieties, Represent. Theory 4 (2000), 97–126. MR 1789463, DOI 10.1090/S1088-4165-00-00069-8
- C. Chevalley, Sur les décompositions cellulaires des espaces $G/B$, Algebraic groups and their generalizations: classical methods (University Park, PA, 1991) Proc. Sympos. Pure Math., vol. 56, Amer. Math. Soc., Providence, RI, 1994, pp. 1–23 (French). With a foreword by Armand Borel. MR 1278698, DOI 10.1090/pspum/056.1/1278698
- Vinay V. Deodhar, On some geometric aspects of Bruhat orderings. I. A finer decomposition of Bruhat cells, Invent. Math. 79 (1985), no. 3, 499–511. MR 782232, DOI 10.1007/BF01388520
- J. J. Corliss, Upper limits to the real roots of a real algebraic equation, Amer. Math. Monthly 46 (1939), 334–338. MR 4, DOI 10.1080/00029890.1939.11998880
- Bertram Kostant and Shrawan Kumar, $T$-equivariant $K$-theory of generalized flag varieties, J. Differential Geom. 32 (1990), no. 2, 549–603. MR 1072919, DOI 10.4310/jdg/1214445320
- Allen Knutson, A Littelmann-type formula for Duistermaat-Heckman measures, Invent. Math. 135 (1999), no. 1, 185–200. MR 1664699, DOI 10.1007/s002220050283 LL V. Lakshmibai and P. Littelmann: Richardson varieties and equivariant $K$-theory, J. Algebra 260 (2003), 230-260.
- V. Lakshmibai and C. S. Seshadri, Geometry of $G/P$. V, J. Algebra 100 (1986), no. 2, 462–557. MR 840589, DOI 10.1016/0021-8693(86)90089-X
- V. Lakshmibai and C. S. Seshadri, Standard monomial theory, Proceedings of the Hyderabad Conference on Algebraic Groups (Hyderabad, 1989) Manoj Prakashan, Madras, 1991, pp. 279–322. MR 1131317
- Peter Littelmann, A Littlewood-Richardson rule for symmetrizable Kac-Moody algebras, Invent. Math. 116 (1994), no. 1-3, 329–346. MR 1253196, DOI 10.1007/BF01231564
- Peter Littelmann, Contracting modules and standard monomial theory for symmetrizable Kac-Moody algebras, J. Amer. Math. Soc. 11 (1998), no. 3, 551–567. MR 1603862, DOI 10.1090/S0894-0347-98-00268-9
- V. B. Mehta and Wilberd van der Kallen, On a Grauert-Riemenschneider vanishing theorem for Frobenius split varieties in characteristic $p$, Invent. Math. 108 (1992), no. 1, 11–13. MR 1156382, DOI 10.1007/BF02100595
- S. Ramanan and A. Ramanathan, Projective normality of flag varieties and Schubert varieties, Invent. Math. 79 (1985), no. 2, 217–224. MR 778124, DOI 10.1007/BF01388970
- A. Ramanathan, Schubert varieties are arithmetically Cohen-Macaulay, Invent. Math. 80 (1985), no. 2, 283–294. MR 788411, DOI 10.1007/BF01388607
- A. Ramanathan, Equations defining Schubert varieties and Frobenius splitting of diagonals, Inst. Hautes Études Sci. Publ. Math. 65 (1987), 61–90. MR 908216, DOI 10.1007/BF02698935
- R. W. Richardson, Intersections of double cosets in algebraic groups, Indag. Math. (N.S.) 3 (1992), no. 1, 69–77. MR 1157520, DOI 10.1016/0019-3577(92)90028-J
Additional 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