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Uspekhi Mat. Nauk, 1999, Volume 54, Issue 5(329), Pages 3–24 (Mi umn202)  

This article is cited in 7 scientific papers (total in 7 papers)

Unitary reflection groups associated with singularities of functions with cyclic symmetry

V. V. Goryunov

University of Liverpool

Abstract: Finite groups generated by Euclidean reflections have been commonplace in various problems of singularity theory since their relationship with the classification of critical points of functions was discovered by Arnol'd [1], [2]. We show that a number of finite groups generated by unitary reflections are also naturally related to singularities of functions, namely, those invariant under a unitary reflection of finite order. To this end, we consider germs of functions on a manifold with boundary and lift them to a cyclic covering of the manifold, ramified over the boundary. This construction provides a new notion of roots for the groups under consideration and provides skew-Hermitian analogues of these groups.

DOI: https://doi.org/10.4213/rm202

Full text: PDF file (353 kB)
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English version:
Russian Mathematical Surveys, 1999, 54:5, 873–893

Bibliographic databases:

UDC: 515.17
MSC: Primary 32S05, 57R70; Secondary 51F15, 20F55, 32S55, 58F36, 58E05
Received: 29.04.1999

Citation: V. V. Goryunov, “Unitary reflection groups associated with singularities of functions with cyclic symmetry”, Uspekhi Mat. Nauk, 54:5(329) (1999), 3–24; Russian Math. Surveys, 54:5 (1999), 873–893

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Goryunov V.V., “Unitary reflection groups and automorphisms of simple hypersurface singularities”, New Developments in Singularity Theory, Nato Science Series, Series II: Mathematics, Physics and Chemistry, 21, 2001, 305–328  mathscinet  zmath  isi
    2. Slodowy P., “Simple singularities and complex reflections”, New Developments in Singularity Theory, Nato Science Series, Series II: Mathematics, Physics and Chemistry, 21, 2001, 329–348  mathscinet  isi
    3. Proc. Steklov Inst. Math., 258 (2007), 44–52  mathnet  crossref  mathscinet  zmath  elib
    4. Dolgachev, IV, “Reflection groups in algebraic geometry”, Bulletin of the American Mathematical Society, 45:1 (2008), 1  crossref  mathscinet  zmath  isi
    5. Proc. Steklov Inst. Math., 267 (2009), 91–103  mathnet  crossref  mathscinet  zmath  isi  elib
    6. V. V. Goryunov, J. A. Haddley, “Invariant Symmetries of Unimodal Function Singularities”, Mosc. Math. J., 12:2 (2012), 313–333  mathnet  mathscinet  zmath
    7. Małgorzata Mikosz, Andrzej Weber, “Triality, characteristic classes,
      $$D_4$$
      D 4 and
      $$G_2$$
      G 2 singularities”, J. Homotopy Relat. Struct, 2014  crossref  mathscinet  isi  scopus  scopus
  • Успехи математических наук Russian Mathematical Surveys
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