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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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