The Riccati flow and singularities of Schubert varieties
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- by James S. Wolper PDF
- Proc. Amer. Math. Soc. 123 (1995), 703-709 Request permission
Abstract:
Let $\operatorname {Gr}(m,n)$ be the Grassmannian of m-dimensional subspaces of an n-dimensional k-vector space, with $k = {\mathbf {R}}$ or C. Fix an $n \times n$ matrix R with coefficients in k. The Riccati Flow $\Phi$ on $\operatorname {Gr}(m,n)$ is the action of a one-parameter subgroup of ${\text {GL}_n}(k)$, given by ${\Phi _t}(\Lambda ) = {e^{Rt}}\Lambda$. We prove: Theorem. Let X be a Schubert variety in $\operatorname {Gr}(m,n)$. Then there exists a Riccati flow $\Phi$ on X and a stable manifold W for $\Phi$ such that W is the smooth locus of X. Corollary (over C). X as above is smooth if and only if the cohomology of X satisfies Poincaré Duality.References
- M. F. Atiyah, Convexity and commuting Hamiltonians, Bull. London Math. Soc. 14 (1982), no. 1, 1–15. MR 642416, DOI 10.1112/blms/14.1.1 R. Brockett, Finite dimensional linear systems, Wiley, New York, 1969.
- A. Białynicki-Birula, Some theorems on actions of algebraic groups, Ann. of Math. (2) 98 (1973), 480–497. MR 366940, DOI 10.2307/1970915
- Vinay V. Deodhar, Local Poincaré duality and nonsingularity of Schubert varieties, Comm. Algebra 13 (1985), no. 6, 1379–1388. MR 788771, DOI 10.1080/00927878508823227
- Sergei Gelfand and Robert MacPherson, Verma modules and Schubert cells: a dictionary, Paul Dubreil and Marie-Paule Malliavin Algebra Seminar, 34th Year (Paris, 1981) Lecture Notes in Math., vol. 924, Springer, Berlin-New York, 1982, pp. 1–50. MR 662251
- Michiel Hazewinkel and Clyde F. Martin, Representations of the symmetric group, the specialization order, systems and Grassmann manifolds, Enseign. Math. (2) 29 (1983), no. 1-2, 53–87. MR 702734
- Robert A. Proctor, Classical Bruhat orders and lexicographic shellability, J. Algebra 77 (1982), no. 1, 104–126. MR 665167, DOI 10.1016/0021-8693(82)90280-0
- Mark A. Shayman, Phase portrait of the matrix Riccati equation, SIAM J. Control Optim. 24 (1986), no. 1, 1–65. MR 818936, DOI 10.1137/0324001
- James S. Wolper, A combinatorial approach to the singularities of Schubert varieties, Adv. Math. 76 (1989), no. 2, 184–193. MR 1013667, DOI 10.1016/0001-8708(89)90048-0
Additional Information
- © Copyright 1995 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 123 (1995), 703-709
- MSC: Primary 14M15; Secondary 58F25
- DOI: https://doi.org/10.1090/S0002-9939-1995-1221729-X
- MathSciNet review: 1221729