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The story of this talk begins with the very classical Hermitian sum
problem, which asks: if the eigenvalues of two Hermitian matrices
are known, what can we say about the possible eigenvalues of their sum?
The answer is, nowadays, quite a bit (though it took over a hundred
years for us to get there). We know that the possible lists of
eigenvalues form a convex polyhedal cone. The integer points in this
cone have an interpretation in representation theory. The facets of
this cone are described via the geometry of Schubert varieties,
which in turn produces a combinatorial description.
Recently, Belkale and Kumar introduced a ring structure, which they
essentially defined by taking a complicated ring (the cohomology ring of
a generalized flag variety), and changing the product so as to ignore the
complicated parts. Amazingly, this simplification led to a number
breakthroughts in generalizing the Hermitian sum picture, including
descriptions of facets for related problems, and information about
higher codimension faces.
My goal in this talk is to give an overview of the mathematics surrounding
the Hermitian sum problem, and to discuss, in particular, the combinatorics
of the Belkale-Kumar product. No knowledge of representation theory,
Schubert varieties, or cohomology will be assumed. This is joint
work (in progress) with Allen Knutson.
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