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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

MSC:

57N80 Stratifications in topological manifolds
57S20 Noncompact Lie groups of transformations
57T15 Homology and cohomology of homogeneous spaces of Lie groups
32Q99 Complex manifolds
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