Coinvariant Algebras of Finite Subgroups of SL(3;C)

Gomi, Yasushi ; Nakamura, Iku ; Shinoda, Ken-ichi

Canadian journal of mathematics, Tome 56 (2004) no. 3, pp. 495-528

Résumé

For most of the finite subgroups of $\text{SL(3,}\,\text{C)}$ we give explicit formulae for the Molien series of the coinvariant algebras, generalizing McKay's formulae $\text{ }\!\![\!\!\text{ McKay99 }\!\!]\!\!\text{ }$ for subgroups of $\text{SU(2)}$ . We also study the $G $ -orbit Hilbert scheme $\text{Hil}{{\text{b}}^{G}}({{\mathbf{C}}^{3}})$ for any finite subgroup $G $ of $\text{SO(3)}$ , 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 $\text{Hil}{{\text{b}}^{G}}({{\mathbf{C}}^{3}})$ 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 $\text{SO(3)}$ version of the McKay correspondence in the $\text{SU(2)}$ case.

Détail

DOI : 10.4153/CJM-2004-023-4

Mots-clés : 14J30, 14J17, Hilbert scheme, Invariant theory, Coinvariant algebra, McKay quiver, McKay correspondence

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