Affine Grassmannians
type: Item
instance of: scholarly article
title: Affine Grassmannians
zbMATH Open document ID: 0823.51014
DOI: 10.1016/0097-3165(95)90094-2
author: Hans Cuypers
published in: Journal of Combinatorial Theory. Series A
publication date: 1995-10-23T00:00:00Z
review text: Let \({\mathbf P \mathbf G}(V)\) be an \((n + 1)\)-dimensional projective space arising from a right \((n + 2)\)-dimensional vector space over some skewfield \(D\). Let \(0 \leq k \leq n\) and let \({\mathcal H}\) be a collection of \(k\)-dimensional subspaces of \({\mathbf P \mathbf G} (V)\) with the property that for every \((k-1)\)-dimensional subspace \(H_ 0\) and every \((k+1)\)- dimensional subspace \(H_ 1\) with \(H_ 0 \subseteq H_ 1\), the set of all \(k\)-dimensional subspaces containing \(H_ 0\) and contained in \(H_ 1\) either contains a unique element of \(\mathcal H\) or is entirely contained in \(\mathcal H\). Thus \(\mathcal H\) is a geometric hyperplane of the \(k\)-Grassmann geometry of \({\mathbf P \mathbf G} (V)\). The author characterizes by the diagram the subgeometry of \({\mathbf P \mathbf G} (V)\) obtained from \({\mathbf P \mathbf G} (V)\) by deleting all subspaces of dimension \(\geq k + 2\) and all those of dimension \(\leq k + 1\) which do not contain and are not contained in any \(k\)-dimensional subspace not belonging to \(\mathcal H\). The diagram is an \(A_{k + 2}\)-diagram in which the \(k\)th edge is replaced by an edge representing a dual affine plane, and the \((k+1)\)th edge (= last edge) is replaced by an edge representing the class of linear spaces. One additional assumption is hypothesized: if \(k > 1\), then lines must have at least 5 points.
Mathematics Subject Classification ID: 51E24
zbMATH DE Number: 753619
zbMATH Keywords: Grassmann geometry
zbMATH Keywords: geometric hyperplanes
reviewed by: Hendrik Van Maldeghem
MaRDI profile type: Publication
full work available at URL: 0097-3165(95)90094-2
OpenAlex ID: W2913868298
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description: scientific article; zbMATH DE number 753619