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Coinvariant Algebras of Finite Subgroups of $\SL(3,C)$

  Published:2004-06-01
 Printed: Jun 2004
  • Yasushi Gomi
  • Iku Nakamura
  • Ken-ichi Shinoda
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Abstract

For most of the finite subgroups of $\SL(3,\mathbf{C})$, we give explicit formulae for the Molien series of the coinvariant algebras, generalizing McKay's formulae \cite{M99} for subgroups of $\SU(2)$. We also study the $G$-orbit Hilbert scheme $\Hilb^G(\mathbf{C}^3)$ for any finite subgroup $G$ of $\SO(3)$, which is known to be a minimal (crepant) resolution of the orbit space $\mathbf{C}^3/G$. In this case the fiber over the origin of the Hilbert-Chow morphism from $\Hilb^G(\mathbf{C}^3)$ to $\mathbf{C}^3/G$ consists of finitely many smooth rational curves, whose planar dual graph is identified with a certain subgraph of the representation graph of $G$. This is an $\SO(3)$ version of the McKay correspondence in the $\SU(2)$ case.
Keywords: Hilbert scheme, Invariant theory, Coinvariant algebra, McKay quiver, McKay correspondence Hilbert scheme, Invariant theory, Coinvariant algebra, McKay quiver, McKay correspondence
MSC Classifications: 14J30, 14J17 show english descriptions $3$-folds [See also 32Q25]
Singularities [See also 14B05, 14E15]
14J30 - $3$-folds [See also 32Q25]
14J17 - Singularities [See also 14B05, 14E15]
 

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