  For any finite poset P we introduce a homogeneous space as a quotient of the
general linear group with the incidence group of P. When P is a chain this
quotient is a flag variety; for the trivial poset our construction gives a
variety recently introduced in [20]. Moreover we provide decompositions for any
set in a projective space, induced by the action of the incidence group of a
suitable poset. In the classical cases of Grassmannians and flag varieties we
recover, depending on the choice of the poset, the partition into Schubert
cells and the matroid strata. Our general framework produces, for the
homogeneous spaces corresponding to the trivial posets, a stratification by
parking functions.
