Work of Klyachko and of Knutson and Tao in the 1990's
established the Horn conjecture, which is a recursively
defined set of inequalities among eigenvalues of hermitian
matrices A, B, and A+B. This used representation theory,
Schubert calculus, and combinatorics. A consequence is that
other problems in mathematics have a similar Horn recursion,
for example when is a Littlewood-Richardson number non-zero?
Its geometric counterpart is to determine when a triple of
Schubert varieties in a Grassmannian must meet. The
partition indices of such a triple of Schubert varieties are
called a feasible triple.
The answer is that a triple is feasible if and only if the
three partitions satisfy Horn inequalities parametrized by
all feasible triples for smaller Grassmannians. In brief,
the Schubert calculus on a Grassmannian is controlled (to
some degree) by the Schubert calculus on all smaller
My first talk will begin by discussing the Horn
inequalities for eigenvalues of hermitian matrices A, B, and
A+B, and what they mean for geometry. Then I will discuss
Belkale's geometric proof of the Horn recursion. This
suggests that other related feasibility problems in
Schubert calculus may have a similar recursive description.
My second talk will describe a generalization of this
obtained in collaboration with Kevin Purbhoo of the
University of British Columbia. We give a completely
different Horn recursion in the Schubert calculus, involving
cominuscule flag varieties. This yields a different set of
valid inequalities for feasibility in Grassmannians, as well
as two different (but equivalent) sets of inequalities for
feasibility in the Lagrangian Grassmannian. In addition to
giving these inequalities, we will also describe the
geometry behind our proof.