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Eigenvalues of Hermitian matrices and equivariant cohomology of Grassmannians

Part of: Algebraic combinatorics Basic linear algebra Special varieties 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 andersond@math.jussieu.fr
Edward Richmond
Affiliation:
Department of Mathematics, University of British Columbia, UBC Room 121, 1984 Mathematics Road, Vancouver, BC, V6T 1Z2, Canada email erichmond@math.ubc.ca,ed.richmond@gmail.com
Alexander Yong
Affiliation:
Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA email ayong@uiuc.edu
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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 

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