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Connectivity properties of the Schur-Horn map for real Grassmannians

Augustin-Liviu Mare

Abstract

To any $V$ in the Grassmannian ${\rm Gr}_k({\mathbb R}^n)$ 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 ${\rm Gr}_k({\mathbb R}^n)\to {\mathbb R}^n$ (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.

Connectivity properties of the Schur-Horn map for real Grassmannians

Abstract

To any in the Grassmannian of -dimensional vector subspaces in one can associate the diagonal entries of the () matrix corresponding to the orthogonal projection of to . One obtains a map (the Schur-Horn map). The main result of this paper is a criterion for pre-images of vectors in 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.
Paper Structure (10 sections, 10 theorems, 103 equations, 1 table)

This paper contains 10 sections, 10 theorems, 103 equations, 1 table.

Table of Contents

  1. Introduction
  2. Height functions on Grassmannians
  3. The Grassmannian ${\rm Gr}_k(\mathbb R^n)$ as a real locus
  4. Proof of the main result
  5. Two examples
  6. Application: connectivity of the space of real tight frames
  7. The case when $d_1= \cdots = d_n$
  8. Other examples: the case when $n$ is even
  9. Other examples: the case when $n$ is odd
  10. Relations with spaces of polygons in Euclidean plane.

Key Result

Theorem 1.1

If $d=(d_1, \ldots, d_n)\in \Delta_{n,k}$ is such that then $\mu^{-1}(d)$ is a connected subspace of ${\rm Gr}_k(\mathbb R^n)$.

Theorems & Definitions (28)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Example 1.4
  • Example 1.5
  • Theorem 2.1
  • Remark 2.2
  • Proposition 2.3
  • proof
  • Remark 2.4
  • ...and 18 more