The combinatorics of quiver representations

Derksen, Harm; Weyman, Jerzy

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

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.

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.

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