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Eigenvalues of Hermitian matrices and equivariant cohomology of Grassmannians

Part of: Special varieties Basic linear algebra Algebraic combinatorics Differential topology

Published online by Cambridge University Press:  07 August 2013

David Anderson ,
Edward Richmond  and
Alexander Yong
David Anderson
Affiliation:
FSMP–Institut de Mathématiques de Jussieu, 75013 Paris, France email [email protected]
Edward Richmond
Affiliation:
Department of Mathematics, University of British Columbia, UBC Room 121, 1984 Mathematics Road, Vancouver, BC, V6T 1Z2, Canada email [email protected],[email protected]
Alexander Yong
Affiliation:
Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA email [email protected]
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Abstract

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The saturation theorem of Knutson and Tao concerns the nonvanishing of Littlewood–Richardson coefficients. In combination with work of Klyachko, it implies Horn’s conjecture about eigenvalues of sums of Hermitian matrices. This eigenvalue problem has a generalization to majorized sums of Hermitian matrices, due to S. Friedland. We further illustrate the common features between these two eigenvalue problems and their connection to Schubert calculus of Grassmannians. Our main result gives a Schubert calculus interpretation of Friedland’s problem, via equivariant cohomology of Grassmannians. In particular, we prove a saturation theorem for this setting. Our arguments employ the aforementioned work together with recent work of H. Thomas and A. Yong.

Keywords

eigenvalue problemequivariant cohomologySchubert calculus

MSC classification

Secondary: 15A42: Inequalities involving eigenvalues and eigenvectors 14M15: Grassmannians, Schubert varieties, flag manifolds 05E15: Combinatorial aspects of groups and algebras 57R91: Equivariant algebraic topology of manifolds
Type
Research Article
Information
Compositio Mathematica , Volume 149 , Issue 9 , September 2013 , pp. 1569 - 1582
Copyright
© The Author(s) 2013 

References

Anderson, D., Positivity in the cohomology of flag bundles (after Graham), Preprint (2007), arXiv:0711.0983.Google Scholar
Belkale, P., Geometric proofs of Horn and saturation conjectures, J. Algebraic Geom. 15 (2006), 133–173.Google Scholar
Belkale, P., Quantum generalization of the Horn conjecture, J. Amer. Math. Soc. 21 (2008), 365–408.Google Scholar
Belkale, P. and Kumar, S., Eigenvalue problem and a new product in cohomology of flag varieties, Invent. Math. 166 (2006), 185–228.Google Scholar
Belkale, P. and Kumar, S., Eigencone, saturation and Horn problems for symplectic and odd orthogonal groups, J. Algebraic Geom. 19 (2010), 199–242.CrossRefGoogle Scholar
Buch, A., A Littlewood–Richardson rule for the K-theory of Grassmannians, Acta Math. 189 (2002), 37–78.Google Scholar
Buch, A., Eigenvalues of Hermitian matrices with positive sum of bounded rank, Linear Algebra Appl. 418 (2006), 480–488.Google Scholar
Chindris, C., Eigenvalues of Hermitian matrices and cones arising from quivers, Inter. Math. Res. Notices (2006), 27; Art. ID 59457.Google Scholar
Derksen, H. and Weyman, J., Semi-invariants of quivers and saturation for Littlewood–Richardson coefficients, J. Amer. Math. Soc. 13 (2000), 456–479.Google Scholar
Friedland, S., Finite and infinite-dimensional generalizations of Klyachko’s theorem, Linear Algebra Appl. 319 (2000), 3–22.Google Scholar
Fulton, W., Eigenvalues of majorized Hermitian matrices and Littlewood–Richardson coefficients, Linear Algebra Appl. 319 (2000), 23–36.Google Scholar
Fulton, W., Eigenvalues, invariant factors, highest weights, and Schubert calculus, Bull. Amer. Math. Soc. (N.S.) 37 (2000), 209–249.Google Scholar
Fulton, W., Equivariant cohomology in algebraic geometry, Lectures at Columbia University, Notes by D. Anderson (2007), http://www.math.washington.edu/~dandersn/eilenberg.Google Scholar
Horn, A., Eigenvalues of sums of Hermitian matrices, Pacific J. Math. 12 (1962), 225–241.Google Scholar
Kapovich, M. and Millson, J. J., A path model for geodesics in Euclidian buildings and its applications to representation theory, Groups Geom. Dyn. 2 (2008), 405–480.Google Scholar
Klyachko, A. A., Stable vector bundles and Hermitian operators, Selecta Math. (N.S.) 4 (1998), 419–445.Google Scholar
Knutson, A. and Tao, T., The honeycomb model of ${\mathrm{GL} }_{n} ( \mathbb{C} )$ tensor products I: proof of the saturation conjecture, J. Amer. Math. Soc. 12 (1999), 1055–1090.Google Scholar
Knutson, A. and Tao, T., Puzzles and (equivariant) cohomology of Grassmannians, Duke Math. J. 119 (2003), 221–260.Google Scholar
Kumar, S., Tensor product decomposition, in Proceedings of the International Congress of Mathematicians, vol. III (Hindustan Book Agency, New Delhi, 2010), 1226–1261.Google Scholar
Molev, A. and Sagan, B., A Littlewood–Richardson rule for factorial Schur functions, Trans. Amer. Math. Soc. 351 (1999), 4429–4443.Google Scholar
Purbhoo, K. and Sottile, F., The recursive nature of cominuscule Schubert calculus, Adv. Math. 217 (2008), 1962–2004.Google Scholar
Ressayre, N., Geometric invariant theory and the generalized eigenvalue problem, Invent. Math. 180 (2010), 389–441.Google Scholar
Sam, S., Symmetric quivers, invariant theory, and saturation theorems for the classical groups, Adv. Math. 229 (2012), 1104–1135.Google Scholar
Thomas, H. and Yong, A., The direct sum map on Grassmannians and jeu de taquin for increasing tableaux, Int. Math. Res. Notices 2011 (2011), 2766–2793.Google Scholar
Thomas, H. and Yong, A., Equivariant Schubert calculus and jeu de taquin, Ann. Inst. Fourier (Grenoble), to appear; arXiv:1207.3209.Google Scholar
Totaro, B., Tensor products of semistables are semistable, in Geometry and analysis on complex manifolds, festschrift for professor S. Kobayashi’s 60th birthday, eds Noguchi, T., Noguchi, J. and Ochiai, T. (World Scientific, Singapore, 1994), 242–250.Google Scholar