Transformations of Grassmannians and automorphisms of classical groups (Q1396038)

The following generalization of the fundamental theorem of projective spaces [see, e.g., \textit{H. Lüneburg}, Arch. Math. {17}, 206--209 (1966; Zbl 0141.18001) or \textit{C. A. Faure} and \textit{A. Frölicher}, Geom. Dedicata {53}, 237--262 (1994; Zbl 0826.51002)] to Grassmannians \({G}_k(V)\) is due to \textit{W. L. Chow} [Ann. Math. (2) 50, 32--67 (1949; Zbl 0040.22901)]: call \(k-\dim S \cap T\) the distance of \(S,T \in {G}_k(V)\). If \(1 < k < n{-}1\), then any distance preserving map of \({G}_k(V)\) is given by a collineation or a sesqui-linear form. The author proves a theorem of a similar nature: given a basis \(B\) of \(V\), there are \(n\choose k\) subspaces of dimension \(k\) which are spanned by vectors of \(B\). These subspaces form a basis set \({\mathcal B}_k\) of the Grassmannian \({G}_k(V)\). If a bijection and its inverse map are from \({\mathcal B}_k\) into itself, then Chow's conclusion holds. This is applied to transformations of the set of \((k,n{-}k)\)-involutions of \(V\) which preserve commutativity.