Coinvariant Algebras of Finite Subgroups of SL(3;C)
Coinvariant Algebras of Finite Subgroups of SL(3;C)
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DOI：
10.4153/cjm-2004-023-4
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发表时间：
2004-06
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影响因子：
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通讯作者：
Yasushi Gomi;I. Nakamura;K. Shinoda
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文献类型：
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作者：
Yasushi Gomi;I. Nakamura;K. Shinoda
Abstract 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.
Abstract 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.