The Horn Recursion for Schur P - and Q - Functions : Extended

A consequence of work of Klyachko and of Knutson-Tao is the Horn recursion to determine when a Littlewood-Richardson coefficient is non-zero. Briefly, a LittlewoodRichardson coefficient is non-zero if and only if it satisfies a collection of Horn inequalities which are indexed by smaller non-zero Littlewood-Richardson coefficients. There are similar Littlewood-Richardson numbers for Schur P and Qfunctions. Using a mixture of combinatorics of root systems, combinatorial linear algebra in Lie algebras, and the geometry of certain cominuscule flag varieties, we give Horn recursions to determine when these other Littlewood-Richardson numbers are non-zero. Our inequalities come from the usual Littlewood-Richardson numbers, and while we give two very different Horn recursions, they have the same sets of solutions. Another combinatorial by-product of this work is a new Horn-type recursion for the usual Littlewood-Richardson coefficients.