Abstract :
This is a continuation on the studies of rook placements and partition varieties in [5]. In this paper, we generalize the partition varieties to a quotient space of certain matrix space module a parabolic subgroup (vs. a Borel subgroup) of a general linear group. We introduce the ideas of γ-compatible partitions, γ-compatible rook placements and γ-compatible rook length polynomials. First we give an explicit formula for the γ-compatible rook length polynomials. Then we construct correspondence between the CW-complex structure of partition varieties in this general setting and the γ-rook placements on a Ferrers board of the shape defined by a γ-compatible partition. We prove that the Poincare polynomials of cohomology for such a partition variety is given by a γ-compatible rook length polynomial. The model of partition varieties in this general setting generalizes Grassmann manifolds and flag manifolds which gives a uniform and combinatorial treatment for the cohomology of Grassmannians and flag manifolds.
