Skew-symmetric Vanishing Lattices and Intersections of Schubert Cells

In the present paper we apply the theory of skew-symmetric vanishing lattices developed around 15 years ago by B. Wajnryb, S. Chmutov, and W. Janssen for the necessities of the singularity theory to the enumeration of connected components in the intersection of two open opposite Schubert cells in the space of complete real flags. Let us briefly recall the main topological problem considered in [SSV] and reduced there to a group-theoretical question solved below. Let N be the group of real unipotent uppertriangular (n+1)× (n + 1) matrices and Di be the determinant of the submatrix formed by the first i rows and the last i columns. Denote by ∆i the divisor {Di = 0} ⊂ N n+1 and let ∆ be the union ∪i=1∆i. Consider now the complement U n+1 = N \∆. The space U can be interpreted as the intersection of two open opposite Schubert cells in SLn+1(R)/B. In [SSV] we have studied the number of connected components in U. The main result of [SSV] can be stated as follows. Consider the vector space T = T(F2) of upper triangular n×n-matrices over F2. We define the group Gn as the subgroup of GL(T ) generated by F2-linear transformations gij , 1 6 i 6 j 6 n − 1. The generator gij acts on a matrix M ∈ T n as follows. Let M ij denote the 2 × 2 submatrix of M formed by rows i and i + 1 and columns j and j + 1 (or its upper triangle in case i = j). Then gij applied to M changes M ij by adding to each entry of M ij the trace of M ij , and does not change all the other entries of M . For example, if i < j, then gij changes M ij as follows: ( mij mi,j+1 mi+1,j mi+1,j+1 )