Combinatorial geometries and strata of a torus on homogeneous compact manifolds.

*(Russian)*Zbl 0629.14035The paper is devoted to the analysis of the partition of a compact homogeneous space of a complex semisimple group into orbits with respect to the action of its Cartan subgroups. The first two sections deal with the case of the Grassmann variety. The authors define strata in three equivalent ways: in terms of the moment map, in terms of projective configurations and as intersections of some Schubert cells. Most results of the first section are also contained in the joint paper of the authors and R. M. Goresky and R. M. MacPherson [Adv. Math. 63, 301–316 (1987; Zbl 0622.57014)]. The second section contains discussion of matroids and strata on Grassmannians. The remaining sections are devoted to an extension of some previous results to the general case of a homogeneous space \(G/P\) where \(G\) is a complex semisimple group and \(P\) is its parabolic subgroup. In the last section some examples are discussed.

Reviewer: T.JozĂ©fiak