Connectivity properties of the Schur-Horn map for real Grassmannians

2023/09/20 by Mare, Augustin-Liviu · 2 citations
#14M15 #53C42 #53D20 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2309.11596

Abstract

To any V in the Grassmannian \rm Gr_k(\mathbb Rⁿ) of k-dimensional vector subspaces in \mathbb Rⁿ one can associate the diagonal entries of the (n× n) matrix corresponding to the orthogonal projection of \mathbb Rⁿ to V. One obtains a map \rm Gr_k(\mathbb Rⁿ)→ \mathbb Rⁿ (the Schur-Horn map). The main result of this paper is a criterion for pre-images of vectors in \mathbb Rⁿ 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.

Cited by

• Simply connectedness of spaces of tight frames
• Optimization and the Topology of Spaces of Parseval Frames

Related

• K-theory of real Grassmann manifolds
• The Bruhat decomposition of real Grassmann manifolds
• On construction of k-regular maps to Grassmannians via algebras of socle dimension two
• The manifold of finite rank projections in the space L(H)
• Subspaces with or without a common complement