The combinatorics of quiver representations

Derksen, Harm; Weyman, Jerzy

Annales de l'Institut Fourier, Volume 61 (2011) no. 3, pp. 1061-1131.

We give a description of faces, of all codimensions, for the cones spanned by the set of weights associated to the rings of semi-invariants of quivers. For a triple flag quiver and its faces of codimension 1 this description 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 an Appendix we include a variant of Belkale’s geometric proof of a conjecture of Fulton that works for arbitrary quivers.

On donne une description des faces, des toutes codimensions, pour les cônes engendrés par l’ensemble des poids associés aux anneaux des semi-invariants des carquois. Pour un carquois de drapeaux triples et ses faces de codimension 1, la description est équivalente à un résultat de Knutson-Tao-Woodward sur les facettes du cône de Klyachko. On donne des nouvelles applications aux coefficients de Littlewood-Richardson, en particulier une formule pour les coefficients qui correspond à des triples de partitions sur un mur du cône de Klyachko. On commence par rappeler les méthodes utilisées (suites de Schur, les suites exceptionnelles, les catégories orthogonaux, les décompositions semi-stables, et les quotients GIT pour les carquois). Dans une appendice, on donne une variante d’une démonstration géométrique de Belkale d’une conjecture de Fulton qui est valable pour un carquois quelconque.

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