Rigid ternary relations in finite-dimensional Hilbert-space Grassmannians

Abstract

For positive integers $1\le r<d<n$ consider subsets $S\subseteq \mathbb{G}(r,V)$ of the $r$-plane Grassmannian of an $n$-dimensional Hilbert space $V$ saturated in the sense that the $r$-plane $η&#39;&#39;$ belongs to $S$ whenever it is the orthogonal projection of $η&#39;\in S$ onto a $d$-plane through $η\in S$. Motivated by such closure operators&#39; natural occurrence in projective-geometry and linear preserver problems, we classify said saturated sets as precisely the disjoint unions of Grassmannian spines, with cores standing in a relation of mutual separation that can be made precise (a spine being the set of $r$-planes containing a fixed core $k$-plane $π$ for some $0\le k\le r$). This generalizes the author&#39;s results describing saturated $r$-plane sets in the tame dimensional regime $2r\le d$, where the disjoint unions in question by necessity collapse to single spines.