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 Hilb_^Sₙ (Cⁿ) parametrizing symmetric subschemes of Cⁿ whose coordinate rings, as Sₙ-modules, are isomorphic to a given representation. In the case that = M^ is a permutation module corresponding to certain special types of partitions of n, we prove that Hilb_^Sₙ (Cⁿ) is irreducible or even smooth. We also prove irreducibility whenever 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 Hilb_^Sₙ (Cⁿ). A central tool is the combinatorial theory of higher Specht polynomials.