Symmetric ideals and invariant Hilbert schemes

A symmetric ideal is an ideal in a polynomial ring which is stable under all permutations of the variables. In this paper we initiate a global study of zero-dimensional symmetric ideals. By this we mean a geometric study of the invariant Hilbert schemes ${\mathrm{Hilb}}_{\rho }^{S_n}\left(C^n\right)$ parametrizing symmetric subschemes of $C^n$ whose coordinate rings, as $S_n$-modules, are isomorphic to a given representation ρ. In the case that $\rho =M^{\lambda }$ is a permutation module corresponding to certain special types of partitions λ of n, we prove that ${\mathrm{Hilb}}_{\rho }^{S_n}\left(C^n\right)$ is irreducible or even smooth. We also prove irreducibility whenever $\dim \rho \le 2n$ and the invariant Hilbert scheme is non-empty. In this same range, we classify all homogeneous symmetric ideals and decide which of these define singular points of ${\mathrm{Hilb}}_{\rho }^{S_n}\left(C^n\right)$. A central tool is the combinatorial theory of higher Specht polynomials.