Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

McKay quivers and extended Dynkin diagrams


Authors: Maurice Auslander and Idun Reiten
Journal: Trans. Amer. Math. Soc. 293 (1986), 293-301
MSC: Primary 16A64; Secondary 20C15
DOI: https://doi.org/10.1090/S0002-9947-1986-0814923-X
MathSciNet review: 814923
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ k$ be an algebraically closed field and $ G$ a finite nontrivial group whose order is not divisible by the characteristic of $ k$. Associated with an $ m$-dimensional representation of $ G$ is the McKay quiver, whose vertices correspond to the irreducible representations of $ G$. We show that if $ m = 2$, then the underlying graph of the separated McKay quiver is a finite union of extended Dynkin diagrams.


References [Enhancements On Off] (What's this?)

  • [1] M. Auslander, Rational singularities and almost split sequences, Trans. Amer. Math. Soc. (to appear). MR 816307 (87e:16073)
  • [2] M. Auslander, M. Platzeck and I. Reiten, Periodic modules over weakly symmetric algebras, J. Pure Appl. Algebra 11 (1977), 279-291. MR 0472916 (57:12600)
  • [3] E. Brieskorn, Rationale Singularitäten komplexer Flächen, Invent. Math. 4 (1968), 336-358. MR 0222084 (36:5136)
  • [4] V. Dlab and C. Ringel, Indecomposable representations of graphs and algebras, Mem. Amer. Math. Soc. No. 173 (1976). MR 0447344 (56:5657)
  • [5] L. Dornhoff, Group representation theory, Dekker, New York, 1971. MR 0347959 (50:458a)
  • [6] G. Gonzalez-Sprinberg and J. Verdier, Construction géométrique de la correspondance de McKay, Ann. Sci. École Norm. Sup. (4) 16 (1983), 409-449. MR 740077 (85k:14019)
  • [7] D. Happel, U. Preisel and C. Ringel, Binary polyhedral groups and Euclidean diagrams, Manuscripta Math. 31 (1980), 317-329. MR 576503 (82g:20019)
  • [8] H. Knörrer, Untitled preprint.
  • [9] H. Lenzing, Polyhedral groups and the geometric study of tame hereditary algebras (preprint).
  • [10] J. McKay, Graphs, singularities and finite groups, Proc. Sympos. Pure Math., vol. 37, Amer. Math. Soc., Providence, R. I., 1980, pp. 183-186. MR 604577 (82e:20014)
  • [11] I. Reiten and C. Riedtmann, Skew group algebras in the representation theory of artin algebras, J. Algebra 92 (1985). MR 772481 (86k:16024)
  • [12] C. Riedtmann, Algebren, Darstellungsköcher, Überlagerungen und zurück, Comment. Math. Helv. 55 (1980), 199-224. MR 576602 (82k:16039)
  • [13] R. Steinberg, Subgroups of $ S{U_2}$ and Dynkin diagrams, Pacific J. Math. 118 (1985), 587-598. MR 789195 (86g:20016)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 16A64, 20C15

Retrieve articles in all journals with MSC: 16A64, 20C15


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1986-0814923-X
Article copyright: © Copyright 1986 American Mathematical Society

American Mathematical Society