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CONNECTIVITY PROPERTIES OF MCKAY QUIVERS

Part of: Representation theory of groups Graph theory

Published online by Cambridge University Press:  02 October 2020

HAZEL BROWNE Open the ORCID record for HAZEL BROWNE [Opens in a new window]
HAZEL BROWNE*
Affiliation:
School of Mathematics and Statistics, The University of Sydney, Camperdown, New South Wales 2006, Australia
*
e-mail: [email protected]
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Abstract

We present several results on the connectivity of McKay quivers of finite-dimensional complex representations of finite groups, with no restriction on the faithfulness or self-duality of the representations. We give examples of McKay quivers, as well as quivers that cannot arise as McKay quivers, and discuss a necessary and sufficient condition for two finite groups to share a connected McKay quiver.

Keywords

McKay quiverscomplex representations of finite groupsconnectivity of quivers

MSC classification

Primary: 05C25: Graphs and abstract algebra (groups, rings, fields, etc.)
Secondary: 20C05: Group rings of finite groups and their modules 20C15: Ordinary representations and characters
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 103 , Issue 2 , April 2021 , pp. 182 - 194
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Copyright
© 2020 Australian Mathematical Publishing Association Inc.

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References

Auslander, M. and Reiten, I., ‘McKay quivers and extended Dynkin diagrams’, Trans. Amer. Math. Soc. 293(1) (1986), 293–301.CrossRefGoogle Scholar
Burnside, W., Theory of Groups of Finite Order, 2nd edn (Dover Publications, New York, 1955).Google Scholar
Butin, F., ‘Branching law for the finite subgroups of ${\mathbf{SL}}_4\mathbb{C}$ and the related generalized Poincaré polynomials’, Ukrainian Math. J. 67(10) (2016), 1484–1497. Reprint of Ukraïn. Mat. Zh. 67(10) (2015), 1321–1332.CrossRefGoogle Scholar
Butin, F. and Perets, G. S., ‘Branching law for finite subgroups of ${\mathbf{SL}}_3\mathbb{C}$ and McKay correspondence’, J. Group Theory 17(2) (2014), 191–251.CrossRefGoogle Scholar
Clifford, A. H., ‘Representations induced in an invariant subgroup’, Ann. Math. 38(3) (1937), 533–550.CrossRefGoogle Scholar
Du Val, P., ‘On isolated singularities of surfaces which do not affect the conditions of adjunction (part I)’, Math. Proc. Cambridge Philos. Soc. 30(4) (1934), 453–459.10.1017/S030500410001269XCrossRefGoogle Scholar
Gonzalez-Sprinberg, G. and Verdier, J.-L., ‘Construction géométrique de la correspondance de McKay’, Ann. Sci. Éc. Norm. Supér. (4) 16(3) (1983), 409–449.CrossRefGoogle Scholar
Happel, D., Preiser, U. and Ringel, C. M., ‘Binary polyhedral groups and Euclidean diagrams’, Manuscripta Math. 31(1–3) (1980), 317–329.CrossRefGoogle Scholar
Klein, F., Lectures on the Ikosahedron, translated from German by Morrice, George Gavin (Trübner & Co, Ludgate Hill, London, 1888).Google Scholar
McKay, J., ‘Graphs, singularities, and finite groups’, The Santa Cruz Conference on Finite Groups (University of California, Santa Cruz, CA, 1979), Proceedings of Symposia in Pure Mathematics, 37 (American Mathematical Society, Providence, RI 1980), 183–186.Google Scholar
Steinberg, R., ‘Finite subgroups of ${\mathrm{SU}}_2$ , Dynkin diagrams and affine Coxeter elements’, Pacific J. Math. 118(2) (1985), 587–598.CrossRefGoogle Scholar