Top stars and isomorphisms of Grassmann graphs
type: Item
instance of: scholarly article
title: Top stars and isomorphisms of Grassmann graphs
zbMATH Open document ID: 1333.51013
author: Lucijan Plevnik
published in: Beiträge zur Algebra und Geometrie
publication date: 2015-10-16T00:00:00Z
review text: The author gives an analogue of a theorem of \textit{W.-L. Chow} in [Ann. Math. (2) 50, 32--67 (1949; Zbl 0040.22901)] for infinite dimensional vector spaces. He considers vector spaces \(X\) over division rings, the lattice \({\mathcal G}(X)\) of vector subspaces, and the graph \(\Gamma(X)\), whose set of vertices is \({\mathcal G}(X)\) with pairs \((U,V)\) of adjacent subspaces \(U\), \(V\) as edges; \(U\) is called adjacent to \(V\) (\(U\sim V\)) iff \(\dim U /(U\cap V) = \dim V / (U\cap V)\). \({\mathcal C} \subset {\mathcal G}(X)\) is called a component iff \({\mathcal C}\) induces a connected component in the graph \(\Gamma(X)\). For components \({\mathcal C} \subset {\mathcal G}(X)\) and \({\mathcal D} \subset {\mathcal G}(Y)\), where \(X\) and \(Y\) are vector spaces, a bijective map \(\Phi : {\mathcal C} \longrightarrow {\mathcal D}\) which preserves \(\sim\) in both directions is called isomorphism. First, the author gives a careful description of components \({\mathcal C}\) and the sublattices \({\mathcal C}_\pm\) generated by them in \({\mathcal G}(X)\). The main result of the paper describes isomorphisms in the case of infinite diameter of the connected components. First, it is shown that for diameters \(\geq 2\) any isomorphism \(\Phi : {\mathcal C} \longrightarrow {\mathcal D}\) of components may be uniquely extended to an isomorphism or antiisomorphism \(\rho : {\mathcal C}_\pm \longrightarrow {\mathcal D}_\pm\) of lattices. Furthermore, any such \(\rho\) which maps \({\mathcal C}\) bijectively onto \({\mathcal D}\) restricts to an isomorphism of components. Second, for components of infinite diameter such lattice isomorphisms or antiisomorphisms are characterized by two bijective semilinear maps between certain subspaces of \(X\) and \(Y\). Let \({\mathcal G}_{\alpha,\beta}(X)\) denote the set of subspaces of dimension \(\alpha\) and of codimension \(\beta\) such that \(\alpha + \beta = \dim X\). Chow's theorem shows in the finite dimensional case t…
reviewed by: Horst Szambien
Mathematics Subject Classification ID: 51M35
Mathematics Subject Classification ID: 05C12
Mathematics Subject Classification ID: 05C60
Mathematics Subject Classification ID: 14M15
Mathematics Subject Classification ID: 15A04
Mathematics Subject Classification ID: 15A86
Mathematics Subject Classification ID: 51A10
Mathematics Subject Classification ID: 51A45
zbMATH DE Number: 6495595
zbMATH Keywords: adjacent subspaces
zbMATH Keywords: Grassmann graph
zbMATH Keywords: complementary subspaces
zbMATH Keywords: distant graph
zbMATH Keywords: semilinear mapping
MaRDI profile type: Publication
full work available at URL: s13366-014-0227-1
OpenAlex ID: W2003260103
cites work: Q5817043
cites work: On bijections that preserve complementarity of subspaces
cites work: On the geometry of algebraic homogeneous spaces
cites work: The connected components on the projective line over a ring
cites work: Q4040889
cites work: Graphs, defined by Weyl distance or incidence, that determine a vector space
cites work: An elementary proof of the fundamental theorem of projective geometry
cites work: Chow's theorem for linear spaces
cites work: TRANSFORMATIONS ON THE PRODUCT OF GRASSMANN SPACES
cites work: Adjacency preserving transformations of Grassmann spaces
cites work: Q5822102
cites work: Distance preserving mappings of Grassmann graphs
cites work: Surjections on Grassmannians preserving pairs of elements with bounded distance
cites work: Base subsets of Grassmannians: infinite dimensional case
cites work: Embeddings of Grassmann graphs
cites work: Grassmannians of Classical Buildings
cites work: Maps preserving complementarity of closed subspaces of a Hilbert space
DOI: 10.1007/S13366-014-0227-1
description: scientific article; zbMATH DE number 6495595