Nesting maps of Grassmannians (Q2504807)

Summary: Let \(F\) be a field and \(i < j\) be integers between \(1\) and \(n\). A map of Grassmannians \(f : Gr(i, F^n) \to Gr(j, F^n)\) is called nesting, if \(l\) is contained in \(f(l)\) for every \(l\) in \(Gr(i, F^n)\). We show that there are no continuous nesting maps over \(\mathbb{C}\) and no algebraic nesting maps over any algebraically closed field \(F\), except for a few obvious ones. The continuous case is due to Stong and Grover-Homer-Stong; the algebraic case in characteristic zero can also be deduced from their results. In this paper we give new proofs that work in arbitrary characteristic. As a corollary, we give a description of the algebraic subbundles of the tangent bundle to the projective space \(\mathbb{P}^n\) over \(F\). Another application can be found in a recent paper of \textit{G. M. Bergman} [Transform. Groups 11, No. 1, 7--15 (2006; Zbl 1109.15005)].