The Horn recursion in the cominuscule Schubert calculus

 Frank Sottile

   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.

   This talk will discuss joint work with Kevin Purbhoo 
extending this Horn recursion to the Schubert calculus for 
all cominuscule flag varieties, which are analogs of 
Grassmannians for other reductive groups.   Our goal will be
to describe the geometric version of the Horn problem and 
outline the source of our inequalities and the scheme 
of our proof.