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Coinvariant algebras of finite subgroups of \(\text{SL} (3,\mathbb{C})\). (English) Zbl 1066.14053

Let \(G\) be a finite subgroup of \(\text{SL}(n, \mathbb C)\), \(S_G\) the covariant algebra of \(G\) and \(\langle S_G\rangle_i\) the subspace of \(S_G\) of homogeneous degree \(i\). For each irreducible representation \(\rho\) of \(G\) let \(\langle \rho, (S_G)_i\rangle _G\) be the multiplicity of \(\rho\) in \((S_G)_i.\) The Molian series \(P_{S_{G, \rho}} (t)\) of \(S_G\) for \(\rho\) is defined by \[ P_{S_{G, \rho}}(t)=\sum\langle \rho, (S_G)_i\rangle_G t^i. \] Explicit formulas for the Molian series are given in the case that \(G\) is one of the exceptional finite subgroups of \(\text{SL}(3, \mathbb C)\). Let \(\text{Hilb}^G(\mathbb C^n)\) be the universal subscheme of the Hilbert scheme \(\text{Hilb}^{| G| } (\mathbb C^n)\) parameterizing all smoothable scheme theoretic \(G\)-orbits of length \(| G| \). \(\text{Hilb}^G(\mathbb C^3)\) is studied, especially the fiber \(\pi^{-1}(0)\) of the Hilbert-Chow morphism \(\pi: \text{Hilb}^G(\mathbb C^3) \to \mathbb C^3/G\) in case that \(G\) is a finite subgroup of SO(3). An SO(3)-version of the McKay correspondence similar to the SU(2) case is given.

MSC:

14L30 Group actions on varieties or schemes (quotients)
34B05 Linear boundary value problems for ordinary differential equations
14C05 Parametrization (Chow and Hilbert schemes)
14J17 Singularities of surfaces or higher-dimensional varieties
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