Rings of invariants for representations of quivers

Mihai Halic; Mihai-Sorin Stupariu

doi:10.1016/j.crma.2004.12.012

Algebraic Geometry

Rings of invariants for representations of quivers

Presented by: Jean-Pierre Demailly

Mihai Halic ¹; Mihai-Sorin Stupariu ¹

¹ Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland

Comptes Rendus. Mathématique, Volume 340 (2005) no. 2, pp. 135-140.

Abstract (Original language)
Résumé

In this Note we compute the generators of the ring of invariants for quiver factorization problems, generalizing results of Le Bruyn and Procesi. In particular, we find a necessary and sufficient combinatorial criterion for the projectivity of the associated invariant quotients. Further, we show that the non-projective quotients admit open immersions into projective varieties, which still arise from suitable quiver factorization problems.

Dans cette Note nous calculons les générateurs des anneaux d'invariants pour certains problèmes de factorisation associés aux représentations de carquois, généralisant un résultat démontré par Le Bruyn et Procesi. En particulier, nous déduisons un critère combinatoire nécéssaire et suffisant pour la projectivité du quotient. En plus, nous démontrons que les quotients non-projectifs peuvent être immergés de manière ouverte dans varietés projectives qui proviennent elles mêmes de problèmes de factorisation de carquois appropriés.

Received: 2004-06-04
Accepted: 2004-12-07
Published online: 2004-12-30

DOI: 10.1016/j.crma.2004.12.012

Author's affiliations:

Mihai Halic ¹; Mihai-Sorin Stupariu ¹

¹ Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland

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Mihai Halic; Mihai-Sorin Stupariu. Rings of invariants for representations of quivers. Comptes Rendus. Mathématique, Volume 340 (2005) no. 2, pp. 135-140. doi : 10.1016/j.crma.2004.12.012. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.12.012/

References
Cited by

[1] L. Le Bruyn; C. Procesi Semisimple representations of quivers, Trans. Amer. Math. Soc., Volume 317 (1990), pp. 585-598

[2] S.K. Donaldson Instantons and geometric invariant theory, Commun. Math. Phys., Volume 93 (1984), pp. 453-460

[3] U. Helmke A compactification of the space of rational transfer functions by singular systems, J. Math. Systems Estim. Control, Volume 3 (1993), pp. 459-472

[4] Ch. Okonek; A. Teleman Gauge theoretical Gromov–Witten invariants and virtual fundamental classes (A. Collino et al., eds.), The Fano Conference, Dipartimento di Matematica, Università di Torino, 2004, pp. 591-623

[5] N. Ressayre The GIT-equivalence for G-line bundles, Geom. Dedicata, Volume 81 (2000), pp. 295-324

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