Rings of Invariants which are Hypersurfaces

  • H. E. A. Eddy CampbellEmail author
  • David L. Wehlau
Part of the Encyclopaedia of Mathematical Sciences book series (EMS, volume 139)


As we have seen, we seek to characterize those representations V of groups G whose rings of invariants are well-behaved. The best behaved rings of invariants, \(\mathbb{K}[V]^{G}\), are those which are polynomial rings, that is, \(\mathbb{K}[V]^{G}\) is generated by dim (V) many invariants. A slightly less well behaved class of examples is provided by those rings of invariants which are hypersurfaces, that is, \(\mathbb{K}[V]^{G}\) is generated by dim (V)+1 many invariants. Those representations with this property have been extensively studied in characteristic 0 by Nakajima (1983). Less is known for modular groups. When G is a Nakajima group with maximal proper subgroup H, the following proposition shows that the ring of H-invariants is a hypersurface (or a polynomial) ring.


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  1. 84.
    —, Rings of invariants of finite groups which are hypersurfaces, J. Algebra 80 (1983), no. 2, 279–294. MR 85e:20036 zbMATHCrossRefMathSciNetGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  1. 1.Sir Howard Douglas Hall, Dept. MathematicsUniversity of New BrunswickFrederictonCanada
  2. 2.Dept. Mathematics & Computer ScienceRoyal Military College of CanadaKingstonCanada

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