Automorphisms of spine spaces
The notion of a spine space has been introduced by the first author in J. Geom. 72, 172-187 (2001; Zbl 1003.51001). Such a space consists of all \(k\)-flats in a projective space \(P\) (of all linear subspaces of a vector space \(V\)) which intersect a given arbitrary subspace \(S\) of \(P\) in an \(m\)-flat. The special case that \(S\) is a hyperplane and \(m=k{-}1\) yields affine Grassmann spaces. Let \(W\) be a fixed subspace of \(V\) and write \({\mathcal F}_{k,m}(W)\) for the set of all \(k\)-dimensional subspaces \(U<V\) such that \(\dim U\cap W = m\). The spine space determined by \(k, m\), and \(W\) is denoted by \({\mathbf A}_{k,m}(W)\). A \(k\)-pencil consists of all \(k\)-subspaces \(U\) of a \((k+1)\)-dimensional subspace \(B\leq V\) containing a given \((k-1)\)-subspace \(H\) of \(B\). The geometry \({\mathbf P}_k\) has the \(k\)-subspaces \(U<V\) as points and the \(k\)-pencils as lines. Theorem. Each automorphism of \({\mathbf A}_{k,m}(W)\) is the restriction to \({\mathcal F}_{k,m}(W)\) of an automorphism of \({\mathbf P}_k\), it is induced either by a semi-linear bijection of \(V\) or by a sesqui-linear form on \(V\).
scientific article; zbMATH DE number 1868566
scientific article