## DIMACS TR: 94-24

## Rook Placements and Partition Varieties P\M

### Author: Kequan Ding

**
ABSTRACT
**

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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