The Horn recursion in the Schubert calculus

 Frank Sottile
 Texas A&M University

   A consequence of Knutson and Tao's proof of the saturation
conjecture is a conjecture of Horn, which implies that the
non-vanishing of Littlewood-Richardson numbers is recursive:
A Littlewood-Richardson number is non-zero if and only if
its partition indices satisfy the Horn inequalities imposed
by all `smaller' non-zero Littlewood-Richardson numbers.
A way to express this Horn Recursion is that non-vanishing
in the Schubert calculus of a Grassmannian is controlled by
non-vanishing in the Schubert calculus of all smaller Grassmannians.

   In joint work with Kevin Purbhoo, we have extended this
Horn recursion to the cominuscule Schubert calculus.
Our results are interesting even for the Grassmannian,
as the inequalities we obtain are different than the Horn
inequalities.  We also obtain two completely different
sets of inequalities for the analogs of Littlewood-Richardson
numbers for Schur P- and Q- functions.

   In this talk, I will first discuss the classican Horn recursion,
which was about the eigenvalues of sums of Hermitian matrices,
and then relate this to the combinatorics of Littlewood-
Richardson numbers from representation theory and Schubert
calculus, explaining the Horn inequalities.  Next, I
wil explain the combinatorial consequences of this work with
Purbhoo, in particular giving the inequalities we obtain.