P-flag spaces and incidence stratifications

For any finite poset P, we introduce a homogeneous space as a quotient of the general linear group. When P is a chain this quotient is a complete flag variety. Moreover, we provide partitions for any set in a projective space, induced by the action of incidence groups of posets. Our general framework allows to deal with several combinatorial and geometric objects, unifying and extending different structures such as Bruhat orders, parking functions and weak orders on matroids. We introduce the notion of P-flag matroid, extending flag matroids.