DIMACS TR: 94-23
Rook Placements and Cellular Decomposition of Partition Varieties
Author: Kequan Ding
ABSTRACT
In this paper, we study the geometric implication of the rook length
polynomials introduced in the author's thesis. We introduce the idea
of partition varieties. These are certain algebraic varieties which
have CW-complex structures. We prove that the cell structure of a
partition variety is in one to one correspondence with the rook
placements on a Ferrers board. This correspondence enables us to
characterize the geometric attachment between a cell and the closure
of another cell combinatorially. The main result of this paper is that
the Poincare polynomial of cohomology for a partition variety is given
by the corresponding rook length polynomial.
This paper serves as the transition of our studies from combinatoriail
aspects to the geometric aspects. To make it user friendly for the
combinatorialists, we give three appendices on the known results on
Grassmann manifolds and flag manifolds which are used frequently. One
appendix is on a technical a lemma on the embedding of manifolds.
Paper available at:
ftp://dimacs.rutgers.edu/pub/dimacs/TechnicalReports/TechReports/1994/94-23.ps
DIMACS Home Page