$G$-Equivariant Hilbert Schemes and Quadratic Forms

Abstract

For a finite abelian group $G$ acting linearly on the plane let $\mathcal{X}$ be the associated quotient stack. We analyze a fine moduli space which parameterizes zero dimensional substacks of $\mathcal{X}$ whose corresponding subschemes of the plane have a given representation type, denoted by $\mathcal{X}^{[\rho]}$. By means of a Grothendieck-Riemann-Roch theorem of To\"en we give explicit representation theoretic formulas of $\dim(\mathcal{X}^{[\rho]})$ for all $\rho \in \operatorname{Rep}(G)$ which are similar to the McKay correspondence. We also give a classification of all such quotient stacks according to an associated quadratic form called the Euler form. For $[\mathbb{A}^2/\mu_n]$ given by the cyclic action $\zeta \cdot (x,y) = (\zeta x, \zeta^{-1} y)$, we give an interpretation of the corresponding Euler form in terms of border strips which may be removed by a diagram, an equational characterization of $n$-core partitions, and the number of significant arrows of a monomial ideal as defined in"Smooth and Irreducible Multigraded Hilbert Schemes"by D. Maclagan and G. Smith.