The combinatorics of quiver representations

Harm Derksen[1]; Jerzy Weyman[2]

  • [1] University of Michigan Department of Mathematics Ann Arbor MI 48109-1043 (USA)
  • [2] Northeastern University Department of Mathematics Boston MA 02115 (USA)

Annales de l’institut Fourier (2011)

  • Volume: 61, Issue: 3, page 1061-1131
  • ISSN: 0373-0956

Abstract

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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.

How to cite

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Derksen, Harm, and Weyman, Jerzy. "The combinatorics of quiver representations." Annales de l’institut Fourier 61.3 (2011): 1061-1131. <http://eudml.org/doc/219793>.

@article{Derksen2011,
abstract = {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.},
affiliation = {University of Michigan Department of Mathematics Ann Arbor MI 48109-1043 (USA); Northeastern University Department of Mathematics Boston MA 02115 (USA)},
author = {Derksen, Harm, Weyman, Jerzy},
journal = {Annales de l’institut Fourier},
keywords = {Quiver representations; Klyachko cone; Littlewood-Richardson coefficients; representations of quivers; rings of semi-invariants},
language = {eng},
number = {3},
pages = {1061-1131},
publisher = {Association des Annales de l’institut Fourier},
title = {The combinatorics of quiver representations},
url = {http://eudml.org/doc/219793},
volume = {61},
year = {2011},
}

TY - JOUR
AU - Derksen, Harm
AU - Weyman, Jerzy
TI - The combinatorics of quiver representations
JO - Annales de l’institut Fourier
PY - 2011
PB - Association des Annales de l’institut Fourier
VL - 61
IS - 3
SP - 1061
EP - 1131
AB - 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.
LA - eng
KW - Quiver representations; Klyachko cone; Littlewood-Richardson coefficients; representations of quivers; rings of semi-invariants
UR - http://eudml.org/doc/219793
ER -

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