DIMACS TR: 94-24

Rook Placements and Partition Varieties P\M

Author: Kequan Ding


This is a continuation on the studies of rook placements and partition varieties. In this paper, we generalize the paptition varieties to a quotient space of certain matrix space module a parabolic subgroup (vs a Borel subgroup) of general linear group. To study the topological structure of this variety, we introduce the ideas of Gamma-compatible partitions and Gamma-compatible rook placements. Then we introduce Gamma-compatible rook length polynomials. First we give an explicit formula for the Gamma-compatible rook length polynomials. Then we give a constructive correspondence between the CW-complex structure of partition varieties in this general setting and the rook placements on a Ferrers board of the shape defined by a Gamma-compatible partition. We prove that the Poincare polynomials of cohomology for such a partition variety is given by a Gamma-compatible rook length polynomial. The model of partition varieties in this general setting generalized Grassmann manifolds and flag manifolds. Thus we get a uniform treatment for the cohomology of Grassmannians and flag manifolds. This model suggests other combinatorial problems on restricted rook placements on a Ferrers board which are not yet explored.

Paper available at: ftp://dimacs.rutgers.edu/pub/dimacs/TechnicalReports/TechReports/1994/94-24.ps
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