Duke Mathematical Journal

Horn's problem, Vinnikov curves, and the hive cone

David E Speyer

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Abstract

A Vinnikov curve is a projective plane curve that can be written in the form det(xX + yY + zZ) = 0 for X, Y, and Z positive definite Hermitian (n × n)-matrices. Given three n-tuples of positive real numbers, α, β, and γ, there exist A, B, and C ∈ GLnℂ with singular values α, β, and γ and ABC = 1 if and only if there is a Vinnikov curve passing through the 3n points $(-1: \alpha_i^2:0)$, $(0:-1:\beta_i^2)$, and $(\gamma_i^2:0:-1)$. Knutson and Tao proved that another equivalent condition for such A, B, and C to exist is that there is a hive (defined within) whose boundary is (log α, log β, log γ). The logarithms of the coefficients of F approximately form such a hive; this leads to a new proof of Knutson and Tao's result. This paper uses no representation theory and essentially no symplectic geometry. In their place, it uses Viro's patchworking method and a topological description of Vinnikov curves.

Article information

Source
Duke Math. J., Volume 127, Number 3 (2005), 395-427.

Dates
First available in Project Euclid: 18 April 2005

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1113847335

Digital Object Identifier
doi:10.1215/S0012-7094-04-12731-0

Mathematical Reviews number (MathSciNet)
MR2132865

Zentralblatt MATH identifier
1069.14037

Subjects
Primary: 15A42: Inequalities involving eigenvalues and eigenvectors
Secondary: 05E15: Combinatorial aspects of groups and algebras [See also 14Nxx, 22E45, 33C80]

Citation

Speyer, David E. Horn's problem, Vinnikov curves, and the hive cone. Duke Math. J. 127 (2005), no. 3, 395--427. doi:10.1215/S0012-7094-04-12731-0. https://projecteuclid.org/euclid.dmj/1113847335


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