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Combinatorial geometries, convex polyhedra, and Schubert cells. (English) Zbl 0622.57014
The authors study the different decompositions of the Grassmannian \(G^ k_{n-k}\) (of (n-k)-planes in \({\mathbb{C}}^ n)\) into strata. The first decomposition is determined by a certain matroid (or combinatorial geometry) of rank k. Using the moment map \(\mu\) : \(G^ k_{n-k}\to {\mathbb{R}}^ n\) another decomposition of \(G^ k_{n-k}\) into strata is obtained as the union of the orbits of \(({\mathbb{C}}^*)^ n\) whose projection under \(\mu\) is a fixed convex polyhedron. The last stratification is the common refinement of the n ! decompositions of \(G^ k_{n-k}\) into Schubert cells. The main result is that all these three stratifications do coincide. The correspondence between the matroids and certain polyhedra which are characterized by a restriction on their vertices and edges is equivalent to the Steiner exchange axiom.
Reviewer: V.Oproiu

57N80 Stratifications in topological manifolds
57S20 Noncompact Lie groups of transformations
57T15 Homology and cohomology of homogeneous spaces of Lie groups
32Q99 Complex manifolds
Full Text: DOI
[1] \scM. F. Atiyah, Convexity and commuting Hamiltonians, Bull. London Math. Soc.\bf14 (982), 1-15. · Zbl 0482.58013
[2] \scA. A. Beilinson, R. D. MacPherson, and V. Schechtman, Notes on motivic cohomology, Duke Math. J., in press. · Zbl 0632.14010
[3] Crapo, H.H; Rota, G.C, On the foundations of combinatorial theory: combinatorial geometries, (1970), MIT Press Cambridge, Mass · Zbl 0216.02101
[4] Danilov, V.I, The geometry of toric varieties, Uspekhi mat. nauk., 33, 85-134, (1978), translated in Russian Math Surveys · Zbl 0425.14013
[5] Gabrielov, A.M; Gelfand, I.M; Losik, M.V; Gabrielov, A.M; Gelfand, I.M; Losik, M.V, Combinatorial calculation of characteristic classes, Funct. anal. appl., Funct. anal. appl., 9, No. 3, 5-26, (1975) · Zbl 0312.57015
[6] Gelfand, I.M, General theory of hypergeometric functions, Dokl., (1986), in press
[7] Gelfand, I.M; Gelfand, S.I, Generalized hypergeometric equations, Dokl., (1986), in press
[8] Gelfand, I.M; MacPherson, R, Geometry in Grassmannians and a generalization of the dilogarithm, Advan. in math., 44, 279-312, (1982) · Zbl 0504.57021
[9] \scI. M. Gelfand and V. Serganova, to appear.
[10] \scI. M. Gelfand and A. Zelevinsky, Hypergeometric functions, Funct. Anal. Appl., in press.
[11] \scM. Goresky and R. MacPherson, On representations of matroids, to appear. · Zbl 0633.14025
[12] Guillemin, V; Sternberg, S, Convexity properties of the moment map, Invent. math., 67, 491-513, (1982) · Zbl 0503.58017
[13] Hilbert, D; Cohn-Vossen, S, Geometry and the imagination, (1965), Chelsea New York · Zbl 0047.38806
[14] \scR. Hain and R. MacPherson, to appear.
[15] Kirwan, F, Cohomology of quotients in symplectic and algebraic geometry, (1984), Princeton Univ. Press Princeton, N.J · Zbl 0553.14020
[16] MacPherson, R.D, The combinatorial formula of gabrielov, Gelfand, and losik for the first pontrjagin class, () · Zbl 0388.57013
[17] Milnor, J; Stasheff, J, Characteristic classes, () · Zbl 0298.57008
[18] Van der Waerden, B.L, Moderne algebra, (1937), Springer-Verlag Berlin · Zbl 0016.33902
[19] Welsh, D.J.A, Matroid theory, (1976), Academic Press New York · Zbl 0343.05002
[20] Whitney, H, On the abstract properties of linear dependence, Amer. J. math., 57, 509-533, (1935) · JFM 61.0073.03
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