$G$-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 Toën we give explicit representation theoretic formulas of $\dim ({\mathcal{X}}^{[\rho ]})$ for all $\rho \in \mathrm{Rep}(G)$ 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.