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 manifold over Cn. The strata are defined by the dimensions of intersections of one space from each flag, 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 Sn when d equals 2 and to the lattice of partitions of a d-set when n equals 2.