The combinatorics of quiver representations

By Harm Derksen, Jerzy Weyman. 2006

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We give a description of faces of all codimensions for the cones of weights of rings of semi-invariants of quivers. For a triple flag quiver and faces of codimension 1 this reduces to the result of Knutson-Tao-Woodward on the facets of the Klyachko cone. We give new applications to Littlewood-Richardson coefficients, including a product formula for LR-coefficients corresponding to triples of partitions lying on a wall of the Klyachko cone. We systematically review and develop the necessary methods (exceptional and Schur sequences, orthogonal categories, semi-stable decompositions, GIT quotients for quivers). In the Appendix we include a version of Belkale's geometric proof of Fulton's conjecture that works for arbitrary quivers.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/624734d5faa67cab501d99d8feb8bf4e19dfc453