The Combinatorics of Quiver Representations

Abstract

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.

63 pages

References in corpus (3)

• Quivers, long exact sequences and Horn type inequalities
• Geometric Proof of a Conjecture of Fulton
• On the number of subrepresentations of a general quiver representation

Cited by in corpus (6)

• Quiver flag varieties and multigraded linear series
• Quivers, long exact sequences and Horn type inequalities
• Counterexamples to Okounkov's Log-Concavity Conjecture
• Orbit semigroups and the representation type of quivers
• On orbit closures for infinite type quivers
• Cluster fans, stability conditions, and domains of semi-invariants