  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
Littlewood-Richardson 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 Q- functions. 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.
