A decomposition of Fl(n)(d) indexed by permutation arrays

We study a decomposition of Fl(n)(d-1), where: Fl(n) denotes the flag manif old over C-n. The strata are defined by the dimensions of intersections of one space from each fag, so for d equal to 2 this is the usual Bruhat cell decomposition, The strata are indexed by "permutation arrays," which are d- dimensional analogs of permutation matrices. We present a partial order on these permutation arrays, specializing: to the Bruhat order on S-n when d e quals 2 and to the lattice of partitions of a d-set when n equals 2. (C) 20 00 Academic Press.