Abstract
We show that the algebras of semi-invariants of a finite connected quiverQ are complete intersections if and only ifQ is of Dynkin or Euclidean type. Moreover, we give a uniform description of the algebras of semi-invariants of Euclidean quivers.
Similar content being viewed by others
References
S. Abeasis, Codimension 1 orbits and semi-invariants for the representations of an oriented graph of type\(\mathbb{A}_n \), Trans. Amer. Math. Soc.282 (1984), 463–485.
K. Akin, D. Buchsbaum, J. Weyman,Schur functors and Schur complexes, Advances Math.44 (1982), 207–277.
И. Н. Бернштейн И. М. Гельфанд,В. А. Пономарё в,функторы Кокстера и теорема Габрцзля ВМН,28 (1973), 2, 19–33. English translation: I. N. Bernstein, I. M. Gelfand, V. A. Ponomarev,Coxeter functors and Gabriel's theorem, Russian Math. Surveys28 (1973), No. 2, 17–32.
G. Boffi,The universal form of the Littlewood-Richardson rule, Advances Math.68 (1988), 64–84.
D. Buchsbaum, D. Eisenbud,Algebra structures for finite free resolutions and some structure theorems for ideals of codimension 3, Amer. J. Math.99 (1977), 447–485.
C. De Concini, D. Eisenbud, C. Procesi,Young diagrams and determinantal varieties, Inventiones Math.56 (1980), 129–165.
V. Dlab, C. M. Ringel,Indecomposable Representations of Graphs and Algebras, Memoirs Amer. Math. Soc.173 (1976).
S. Donkin,Rational Representations of Algebraic Groups, Lecture Notes in Math.1140, Springer, 1985.
S. Donkin,The normality of closures of conjugacy classes of matrices, Inventiones Math.101 (1990), 717–736.
S. Donkin,Invariants of several matrices, Inventiones Math.110 (1992), 389–401.
S. Donkin,Polynomial invariants of representations of quivers, Comment. Math. Helvetici69 (1994), 137–141.
P. Donovan, M. R. Freislich,The Representation Theory of Finite Graphs and Associated Algebras, Carleton Lecture Notes5, Ottawa, 1973.
D. Eisenbud,Commutative Algebra with a View Toward Algebraic Geometry, Graduate Texts in Math.150, Springer, 1994.
W. Fulton,Young Tableaux, London Mathematical Society Student Texts35, Cambridge University Press, 1997.
P. Gabriel,Unzerlegbare Darstellungen I, Manuscripta Math.6 (1972), 71–103.
D. Happel,Relative invariants of quivers of tame type, J.Algebra86 (1984), 315–335.
R. Howe, R. Huang,Projective invariants of four subspaces, Advances Math.118 (1996), 295–336.
J. E. Humpreys,Linear Algebraic Groups, Graduate Texts in Math.21, Springer, 1975. Russian translation:Д. ХамфриЛэнейнче алгебраическце граппы, Наука, М., 1980.
T. Józefiak, P. Pragacz, J. Weyman,Resolutions of determinantal varieties and tensor complexes associated with symmetric and antisymmetric matrices, Astérisque87–88 (1981), 109–189.
V. G. Kac,Infinite root systems, representations of graphs and invariant theory, Inventiones Math.56 (1980), 57–92.
V. G. Kac,Infinite root systems, representations of graphs and invariant theory II, J. Algebra78 (1982), 141–162.
G. Kempken,Eine Darstellung des Köchers \(\tilde{\mathbb{A}}_k \), Bonner Math. Schriften137 (1982), 1–159.
K. Koike, Relative invariants of the polynomial rings over the type\(\mathbb{D}_r \) quivers, Advances Math.105 (1994), 166–189.
H. Kraft,Geometrische Methoden in der Invariantentheorie, Aspekte der Mathematik, Vieweg, 1985. Russian translation: Ш. Крафт Геометриыеские методы е теории инвариантое Мир М., 1987.
H. Kraft, Ch. Riedtmann,Geometry of representations of quivers, in:Representations of Algebras, London Math. Society Lecture Notes Series116, Cambridge Univ. Press, 1986, 109–145.
L. Le Bruyn, C. Procesi,Semisimple representations of quivers, Trans. Amer. Math. Soc.317 (1990), 585–598.
L. Le Bruyn, Y. Teranishi,Matrix invariants and complete intersections, Glasgow Math. J.32 (1990), 227–229.
I. G. Macdonald,Symmetric Functions and Hall Polynomials, Oxford Mathematical Monographs, Clarendon Press, 1979. Russian translation: И. Макдональд Суммемрическце функзи и многочлены Холла Мир М., 1985.
Л. А. Назарова,Представления колчанов бесконечного мипа Изв АН СССР, сер. мат.37 (1973), 752–791. English translation: L. A Nazarova,Representations of quivers of infinite type, Math. USSR-Izv.7 (1993), 749–792.
C. M. Ringel,Representations of K-species and bimodules, J. Algebra41 (1976), 269–302.
C. M. Ringel,The rational invariants of the tame quivers, Inventiones Math.58 (1980), 217–239.
M. Sato, T. Kimura,A classification of irreducible prohomogenoeous vector spaces and their relative invariants, Nagoya J. Math.65 (1977), 1–155.
A. Schofield,Semi-invariants of quivers, J. London Math. Soc.43 (1991), 385–395.
G. W. Schwarz, D. L. Wehlau,Invariants of four subspaces, Ann. Inst., Fourier, Grenoble48 (1998), 667–687.
А. Н. ЗубкобОбобщениемеоремы Размыслова Прочези Алгебра и Логика,35 (1996) 4, 433–457. English translation: A. N. Zubkov,A generalization of the Razmyslov-Procesi theorem, Algebra and Logic35 (1996), No. 4, 241–254.
A. N. Zubkov,Procesi-Razmyslov's theorem for quivers, Fundamental and Applied Mathematics, Moscow University, in press.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Skowroński, A., Weyman, J. The algebras of semi-invariants of quivers. Transformation Groups 5, 361–402 (2000). https://doi.org/10.1007/BF01234798
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF01234798