Abstract: Work in 1990's and 2000's by several authors characterized
possible eigenvalues of a product AB of unitary n by n matrices A,B,
in terms of eigenvalues of A and B by a system of linear inequalities which
have a recursive nature. This way, one gets an "eigen-polyhedron" depending on n.
After reviewing these older works I will focus on the behavior of these polyhedra as n varies: there is a natural "strict" union of these polyhedra (joint with A. Gibney and A. Kazanova). I will describe open problems on the structure of vertices of the eigen-polyhedron, and the resulting structure of this union (inequalities become points in the union) implied by Horn recursion.
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