The Horn recursion in the cominuscule 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.

   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.  In particular, 
we give two very different sets of necessary and sufficient 
inequalities for the non-vanishing of the analogs of 
Littlewood-Richardson numbers for Schur P- and Q- functions.  
Interestingly, the inequalities we obtain for the ordinary 
Littlewood-Richardson numbers are different than the Horn 
inequalities.