Transversal Matroids and Strata on Grassmannians

1. Let Zk~ be the manifold of all nondegenerate complex k × n-matrices, k < n , let Gkn be the Grassmannian of k-dimensional subspaces in C~', and let ~r: Zkn -~ Gk, be the na tura l projection (7~(z) for z E Zk~ is the k-dimensional subspace of C ~ generated by the rows of z). For a matr ix z E Zkn denote by px(z) = pil...ik(z) the minor of z composed by the columns with indices from the set I = { i l , . . . , i k } C {1, . . , n}. A fixed set B of k-element subsets of { 1 , . . . , n } is called a collection. By the stratum S = SB C Z~, associated with a collection B we mean the manifold of all z ~ Zk~ such tha t pi(z) # 0 .'. .: I ~ B . The image s = sB = ~-(S) of the s t r a tum SB is called a stratum on the Grassmannian Gkn (see [2]). Obviously, Gk, = UB sB. If the s t r a tum associated with a collection B is nonempty, then B satisfies the axioms for bases of a matroid (see [7, 8]). This matro id is said to be associated with the s t ra tum. For U C { 1 , . . . , k} × { 1 , . . . , n} denote by Z(U) the submanifold of matr ices z = (zij) e Zk~ such that zij = 0 whenever (i, j ) ~ U.