Rook placements and generalized partition varieties
type: Item
instance of: scholarly article
title: Rook placements and generalized partition varieties
zbMATH Open document ID: 0887.05057
DOI: 10.1016/S0012-365X(96)00287-7
author: Kequan Ding
published in: Discrete Mathematics
publication date: 1998-04-28T00:00:00Z
review text: The paper generalizes the concept of partition varieties introduced in a preceding paper by the same author. For this purpose, \(\gamma\)-compatible partitions \(\lambda\) are introduced, where \(\gamma\) is a composition of some integers. It is shown that a certain quotient space associated to such composition \(\gamma\) and partition \(\lambda\) is a projective variety. Moreover, this generalized partition variety is a CW-complex which can be described combinatorially in terms of \(\gamma\)-compatible rook placements of the Ferrers board of \(\lambda\). Finally, the Poincaré polynomial \(P_{\lambda,\gamma}(q)\) for the cohomology equals \(RL_{\lambda,\gamma}(q^2)\), where \(RL_{\lambda,\gamma}\) is the \(\gamma\)-rook length polynomial. The combinatorial description of generalized partition varieties applies to the homology and cohomology of flag manifolds and Grassmann manifolds.
reviewed by: Markus Fulmek
Mathematics Subject Classification ID: 05E99
Mathematics Subject Classification ID: 14M15
Mathematics Subject Classification ID: 05A17
Mathematics Subject Classification ID: 05A05
zbMATH DE Number: 1109958
zbMATH Keywords: rook length polynomial
zbMATH Keywords: projective variety
zbMATH Keywords: partition variety
zbMATH Keywords: Grassmannian manifold
zbMATH Keywords: flag manifold
zbMATH Keywords: Schubert cells
zbMATH Keywords: Bruhat order
MaRDI profile type: Publication
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full work available at URL: s0012-365x(96)00287-7
OpenAlex ID: W2063281630
description: scientific article; zbMATH DE number 1109958