The Horn recursion for Schur P- and Q- functions: Extended Abstract

Details The Horn recursion for Schur P- and Q- functions: Extended Abstract The Horn recursion for Schur P- and Q- functions: Extended Abstract

2006-03-08

arxiv.org/abs/math/0603180v1

Mathematics

Combinatorics

9 pages, extended abstract for FPSAC06 conference

Scientific paper

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.

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