Connected Components in the Intersection of Two Open Opposite Schubert Cells in SLn

For more than one hundred years, Schubert decomposition of complete and partial flag manifolds and Schubert calculus have been an intertwining point of algebra, geometry, representation theory, and combinatorics. In this paper, we are concerned with the further refinement of Schubert decomposition theory. Our far and ambitious goal is to describe the topology of intersection of two arbitrary Schubert cells. This problem seems to be very important in connection with representation theory, in particular, with calculation of Kazhdan–Lusztig polynomials, which is known to be a very hard problem. As a minor step in this direction, we calculate in this paper the number of connected components in the intersection of two open opposite Schubert cells in the space of real n-dimensional flags. This question was raised in [A], and later, in connection with criteria of total positivity, in [SS]. Other topological characteristics of pairwise intersections of Schubert cells (not necessarily opposite) are considered in our previous papers [SV], [SSV1], [SSV2]. Despite the fact that in this paper we consider only real algebraic varieties, we hope that our approach can give some information about complex geometry as well. The reasons for this hope are as follows. (a) The calculation method is based on the beautiful and very deep (“very algebraic”) results by Berenstein, Fomin, and Zelevinsky [BFZ] on the Lusztig parametrization of the unipotent lower-triangular matrices,which form an open cell in the flag manifold. It looks like the same type of parametrization and cell decompositions might be used