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.