Base subsets of Grassmannians: infinite dimensional case
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.
scientific article; zbMATH DE number 5077683