Base subsets of Grassmannians: infinite dimensional case
type: Item
instance of: scholarly article
title: Base subsets of Grassmannians: infinite dimensional case
zbMATH Open document ID: 1106.51011
published in: European Journal of Combinatorics
publication date: 2006-12-07T00:00:00Z
review text: Let \(V\) be a vector space of dimension \(\aleph\) and let \(\mathcal S\) be the set of proper subspaces of \(V\). The orbits of GL\((V)\) acting on \( \mathcal S\) are called Grassmannians. For \(\aleph < \infty\) the Grassmannians are given by \({\mathcal G}_k = \{ S \in {\mathcal S}\, |\, \text{ dim} S = k \}\), where \(1 \leq k < \aleph\). Let \(\mathcal G\) be a Grassmannian. Each base of \(V\) defines a base subset of \(\mathcal G\) which consists of all elements of \(\mathcal G\) spanned by vectors of this base. A block space \(\mathcal B(G)\) of \(\mathcal G \) can be defined by taking the elements of \(\mathcal G\) as the set of points and the set of base subsets of \(\mathcal G\) as the set of blocks. In a previous paper [J. Geom. 75, 132--150 (2002; Zbl 1035.51013)], the author proved for \(3\leq \aleph < \infty\) and \(1 < k < \aleph - 1\) that each automorphism of \({\mathcal B}({\mathcal G}_{k})\) is induced by a semilinear isomorphism of \(V\) to itself or to its dual. For \(k=1, \aleph-1\) this is essentially the fundamental theorem of projective geometry. For finite dimensions this proof depends on a result of \textit{W. L. Chow} about automorphisms of Grassmannian graphs [Ann. Math. 50, 32--67 (1949; Zbl 0040.22901)]. In the paper under review the infinite dimensional case is investigated. For any cardinal number \(\alpha \leq \aleph\) set \({\mathcal G}_{\alpha} = \{ S \in {\mathcal S}\,|\, \text{ dim} S = \alpha, \text{ codim} S = \aleph\}\) and let \({\mathcal B}_{\alpha}\) be the family of all base subsets of \({\mathcal G}_{\alpha}\). The following theorem is proved. If \(\alpha < \aleph\), then any automorphism of \( {\mathcal B}({\mathcal G}_{\alpha})\) is induced by a semilinear autmorphism of \(V\). The elegant proof of this result does not depend on Chow's result, since the latter does not generalize to Grassmannian graphs of infinite dimensional vector spaces.
Mathematics Subject Classification ID: 51M35
Mathematics Subject Classification ID: 14M15
zbMATH DE Number: 5077683
zbMATH Keywords: base subset
zbMATH Keywords: Grassmannian
zbMATH Keywords: infinite dimensional
author: Mark Pankov
MaRDI profile type: Publication
full work available at URL: j.ejc.2005.10.002
OpenAlex ID: W1983948018
cites work: On opposition in spherical buildings and twin buildings
cites work: Q5817043
cites work: On bijections that preserve complementarity of subspaces
cites work: Q3821200
cites work: Q4764139
cites work: On the geometry of algebraic homogeneous spaces
cites work: A unified approach to a characterization of Grassmann graphs and bilinear forms graphs
cites work: Q4889826
cites work: Adjacency preserving transformations of Grassmann spaces
cites work: On isomorphisms of Grassmann spaces
cites work: A characterization of Grassmann graphs
cites work: A characterization of Grassmann and Johnson graphs
cites work: Transformations of Grassmannians and automorphisms of classical groups
cites work: Transformations of Grassmannians preserving the class of base subsets
cites work: A characterization of geometrical mappings of Grassmann spaces
cites work: Automorphisms of spine spaces
cites work: Buildings of spherical type and finite BN-pairs
DOI: 10.1016/J.EJC.2005.10.002
description: scientific article; zbMATH DE number 5077683