Top stars and isomorphisms of Grassmann graphs
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 that \(\alpha < \beta\) makes \({\mathcal G}_{\alpha,\beta}(X)\) a component, whose automorphisms may be described by bijective semilinear maps on \(X\). From his main result the author derives that an analogous description for infinite dimensional \(X\) and infinite \(\alpha\) and \(\beta\) is not possible. Continuing the work of \textit{A. Blunck} and \textit{H. Havlicek} [Discrete Math. 301, No. 1, 46--56 (2005; Zbl 1083.51001)], the author applies his results to infinite dimensional vector spaces \(X\) and \(Y\), to characterize complementarity preserving bijections of \({\mathcal G}_{\alpha,\alpha}(X)\longrightarrow {\mathcal G}_{\gamma,\gamma}(Y)\) by certain bijective semilinear maps \(X\longrightarrow Y\).
scientific article; zbMATH DE number 6495595
scientific article