Abstract
To any V in the Grassmannian \(\textrm{Gr}_k({\mathbb R}^n)\) 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 \(\textrm{Gr}_k({\mathbb {R}}^n)\rightarrow {\mathbb {R}}^n\) (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 et al. (SIAM J Appl Algebra Geom 1:38–72, 2017).
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Notes
Guest is actually dealing with the complex Grassmannian \(\textrm{Gr}_k({\mathbb {C}}^n)\); however he points out that similar results hold for the real Grassmannian, see [7, Remark, p. 170].
Cf. also Rem. 2.2.
We actually rely on the remark made in [7] on page 170.
Note that \(d\in \Delta _{n,k}\) if and only if \(1-d\in \Delta _{n,n-k}\).
Neither the numbers \(i_1, \ldots , i_{n'}\) nor \(j_1,\ldots , j_{n''}\) are necessarily in increasing order.
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Acknowledgements
I would like to thank Tom Needham and Clayton Shonkwiler for suggesting the topic of the present paper and for a fruitful exchange of ideas, Jost-Hinrich Eschenburg for helpful discussions, and the anonymous referee for carefully reading the manuscript and suggesting numerous improvements.
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Mare, AL. Connectivity properties of the Schur–Horn map for real Grassmannians. Abh. Math. Semin. Univ. Hambg. 94, 33–55 (2024). https://doi.org/10.1007/s12188-024-00277-1
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DOI: https://doi.org/10.1007/s12188-024-00277-1
Keywords
- Real and complex Grassmann manifolds
- Real loci
- Moment maps
- Convexity properties
- Morse theory
- Stiefel manifolds
- Tight frames