Horn's Problem, Vinnikov Curves and the Hive Cone

Details Horn's Problem, Vinnikov Curves and the Hive Cone Horn's Problem, Vinnikov Curves and the Hive Cone

2003-11-25

arxiv.org/abs/math/0311428v1

Mathematics

Algebraic Geometry

Scientific paper

A Vinnikov curve is a projective plane curve which can be written in the form det(xX+yY+zZ)=0 for X, Y and Z positive definite Hermitian n by n matrices. Given three n-tuples of positive real numbers, alpha, beta and gamma, there exist A, B and C in GL_n \CC with singular values alpha, beta and gamma 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 alpha, log beta, log gamma). 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.

Affiliated with

Also associated with

No associations

Rating

Horn's Problem, Vinnikov Curves and the Hive Cone does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Horn's Problem, Vinnikov Curves and the Hive Cone, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Horn's Problem, Vinnikov Curves and the Hive Cone will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-670865