Connectivity properties of the Schur-Horn map for real Grassmannians

Abstract

To any $V$ in the Grassmannian ${\mathrm{Gr}}_k({\mathbb{R}}^n)$ of $k$-dimensional vector subspaces in ${\mathbb{R}}^n$ one can associate the diagonal entries of the ($n\times n$) matrix corresponding to the orthogonal projection of ${\mathbb{R}}^n$ to $V$. One obtains a map ${\mathrm{Gr}}_k({\mathbb{R}}^n)\to {\mathbb{R}}^n$ (the Schur-Horn map). The main result of this paper is a criterion for pre-images of vectors in ${\mathbb{R}}^n$ to be connected. This will allow us to deduce connectivity criteria for a certain class of subspaces of the real Stiefel manifold which arise naturally in frame theory. We extend in this way results of Cahill, Mixon, and Strawn.

25 pages; v2: exposition improved