Connectivity properties of the Schur-Horn map for real Grassmannians. (English) Zbl 07873896
Summary: 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 \(\mathrm{Gr}_k (\mathbb{R}^n) \rightarrow\mathbb{R}^n\) (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).
MSC:
| 14M15 | Grassmannians, Schubert varieties, flag manifolds |
| 53C42 | Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) |
| 53D20 | Momentum maps; symplectic reduction |
Keywords:
real and complex Grassmann manifolds; real loci; moment maps; convexity properties; Morse theory; Stiefel manifolds; tight framesReferences:
| [1] | Atiyah, MF, Convexity and commuting Hamiltonians, Bull. Lond. Math. Soc., 14, 1-15, 1982 · Zbl 0482.58013 · doi:10.1112/blms/14.1.1 |
| [2] | Baird, T.; Heydari, N., Cohomology of quotients in real symplectic geometry, Alg. Geom. Topol., 22, 3249-3276, 2022 · Zbl 1512.53076 · doi:10.2140/agt.2022.22.3249 |
| [3] | Cahill, J.; Mixon, D.; Strawn, N., Connectivity and irreducibility of algebraic varieties of finite unit norm tight frames, SIAM J. Appl. Algebra Geom., 1, 38-72, 2017 · Zbl 1370.42023 · doi:10.1137/16M1068773 |
| [4] | Casazza, PG; Leon, MT, Existence and construction of finite frames with a given frame operator, Int. J. Pure Appl. Math., 63, 149-157, 2010 · Zbl 1225.42021 |
| [5] | Dykema, K.; Strawn, N., Manifold structure of spaces of spherical tight frames, Int. J. Pure Appl. Math., 28, 217-256, 2006 · Zbl 1134.42019 |
| [6] | Duffin, RJ; Schaeffer, AC, A class of nonharmonic Fourier series, Trans. Am. Math. Soc., 72, 341-366, 1952 · Zbl 0049.32401 · doi:10.1090/S0002-9947-1952-0047179-6 |
| [7] | Guest, M.A.: Morse theory in the 1990s, Invitations to geometry and topology, Oxford Graduate Texts in Mathematics, vol. 7, Oxford Univ. Press, Oxford, pp. 146-207 (2002) · Zbl 0996.54503 |
| [8] | Guillemin, V.; Sternberg, S., Convexity properties of the moment mapping, Invent. Math., 67, 491-513, 1982 · Zbl 0503.58017 · doi:10.1007/BF01398933 |
| [9] | Hausmann, J-C; Knutson, A., Polygon spaces and Grassmannians, Enseign. Math., 43, 173-198, 1997 · Zbl 0888.58007 |
| [10] | Hausmann, J-C; Knutson, A., The cohomology ring of polygon spaces, Ann. Inst. Fourier, 48, 281-321, 1998 · Zbl 0903.14019 · doi:10.5802/aif.1619 |
| [11] | Horn, A., Doubly stochastic matrices and the diagonal of a rotation matrix, Am. J. Math., 76, 620-630, 1954 · Zbl 0055.24601 · doi:10.2307/2372705 |
| [12] | Kapovich, M.; Millson, J., The symplectic geometry of polygons in Euclidean space, J. Differ. Geom., 44, 479-513, 1996 · Zbl 0889.58017 · doi:10.4310/jdg/1214459218 |
| [13] | Kirwan, FC, Cohomology of Quotients in Symplectic and Algebraic Geometry, Mathematical Notes, 1984, New Jersey: Princeton University Press, New Jersey · Zbl 0553.14020 |
| [14] | Łojasiewicz, S., Triangulation of semi-analytic sets, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 18, 449-474, 1964 · Zbl 0128.17101 |
| [15] | Mallat, S.: A wavelet tour of signal processing. The sparse way, 3rd edition, with contributions from Gabriel Peyré, Elsevier/Academic Press, Amsterdam (2009) · Zbl 1170.94003 |
| [16] | Mare, A-L, Connectivity and Kirwan surjectivity for isoparametric submanifolds, Int. Math. Res. Not., 55, 3427-3443, 2005 · Zbl 1097.53056 · doi:10.1155/IMRN.2005.3427 |
| [17] | Marshall, AW; Olkin, I.; Arnold, BC, Inequalities: Theory of Majorization and Its Applications, Springer Series in Statistics, 2011, New York: Springer, New York · Zbl 1219.26003 · doi:10.1007/978-0-387-68276-1 |
| [18] | Mirsky, L., Matrices with prescribed characteristic roots and diagonal elements, J. Lond. Math. Soc., 33, 14-21, 1958 · Zbl 0101.25303 · doi:10.1112/jlms/s1-33.1.14 |
| [19] | Needham, T.; Shonkwiler, C., Symplectic geometry and connectivity of spaces of frames, Adv. Comput. Math., 47, 1, 5, 2021 · Zbl 1464.42028 · doi:10.1007/s10444-020-09842-7 |
| [20] | Needham, T.; Shonkwiler, C., Admissibility and frame homotopy for quaternionic frames, Linear Algebra Appl., 645, 237-255, 2022 · Zbl 1492.42037 · doi:10.1016/j.laa.2022.03.023 |
| [21] | Schur, I., Über eine Klasse von Mittelbildungen mit Anwendungen auf die Determinantentheorie, Sitzungsber. Berl. Math. Ges., 22, 9-20, 1923 · JFM 49.0054.01 |
| [22] | Strohmer, T., Approximation of dual Gabor frames, window decay, and wireless communications, Appl. Comput. Harmon. Anal., 11, 243-262, 2001 · Zbl 0986.42018 · doi:10.1006/acha.2001.0357 |
| [23] | Strohmer, T.; Heath, RW, Grassmannian frames with applications to coding and communication, Appl. Comput. Harmon. Anal., 14, 257-275, 2003 · Zbl 1028.42020 · doi:10.1016/S1063-5203(03)00023-X |
| [24] | Waldron, S., An Introduction to Finite Tight Frames, Applied and Numerical Harmonic Analysis, 2018, New York: Birkhäuser/Springer, New York · Zbl 1388.42078 |
| [25] | Wong, Y-C, A class of Schubert varieties, J. Differ. Geom., 4, 37-51, 1970 · Zbl 0193.49801 · doi:10.4310/jdg/1214429274 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
