DIMACS TR: 94-23

Rook Placements and Cellular Decomposition of Partition Varieties

Author: Kequan Ding


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

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