A geometric approach to alternating $k$-linear forms

Abstract

Given an $n$-dimensional vector space $V$ over a field $\mathbb{K}$, let $2\le k<n$. There is a natural correspondence between the alternating $k$-linear forms $\phi$ of $V$ and the linear functionals $f$ of ${\bigwedge }^{k}V$. Let ${\epsilon }_k:{\mathcal{G}}_k(V)\to \mathrm{PG}({\bigwedge }^{k}V)$ be the Plucker embedding of the $k$-Grassmannian ${\mathcal{G}}_k(V)$ of $V$. Then ${\epsilon }_k^{-1}(\ker (f)\cap {\epsilon }_k({\mathcal{G}}_k(V)))$ is a hyperplane of the point-line geometry ${\mathcal{G}}_k(V)$. All hyperplanes of ${\mathcal{G}}_k(V)$ can be obtained in this way. For a hyperplane $H$ of ${\mathcal{G}}_k(V)$, let $R^{\uparrow }(H)$ be the subspace of ${\mathcal{G}}_{k-1}(V)$ formed by the $(k-1)$-subspaces $A\subset V$ such that $H$ contains all $k$-subspaces that contain $A$. In other words, if $\phi$ is the (unique modulo a scalar) alternating $k$-linear form defining $H$, then the elements of $R^{\uparrow }(H)$ are the $(k-1)$-subspaces $A=⟨a_1,\dots ,a_{k-1}⟩$ of $V$ such that $\phi (a_1,\dots ,a_{k-1},x)=0$ for all $x\in V$. When $n-k$ is even it might be that $R^{\uparrow }(H)=\varnothing$. When $n-k$ is odd, then $R^{\uparrow }(H)\ne \varnothing$, since every $(k-2)$-subspace of $V$ is contained in at least one member of $R^{\uparrow }(H)$. If every $(k-2)$-subspace of $V$ is contained in precisely one member of $R^{\uparrow }(H)$ we say that $R^{\uparrow }(H)$ is spread-like. In this paper we obtain some results on $R^{\uparrow }(H)$ which answer some open questions from the literature and suggest the conjecture that, if $n-k$ is even and at least $4$, then $R^{\uparrow }(H)\ne \varnothing$ but for one exception with $\mathbb{K}\le \mathbb{R}$ and $(n,k)=(7,3)$, while if $n-k$ is odd and at least $5$ then $R^{\uparrow }(H)$ is never spread-like.

29 Pages

References in corpus (1)

• Geometries arising from trilinear forms on low-dimensional vector spaces

Cited by in corpus (1)

• Geometries arising from trilinear forms on low-dimensional vector spaces