Symmetric Ideals and Invariant Hilbert Schemes

Abstract

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 ${\mathrm{Hilb}}_{\rho }^{S_n}({\mathbb{C}}^n)$ parametrizing symmetric subschemes of ${\mathbb{C}}^n$ whose coordinate rings, as $S_n$-modules, are isomorphic to a given representation $\rho$. In the case that $\rho =M^{\lambda }$ is a permutation module corresponding to certain special types of partitions $\lambda$ of $n$, we prove that ${\mathrm{Hilb}}_{\rho }^{S_n}({\mathbb{C}}^n)$ 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 ${\mathrm{Hilb}}_{\rho }^{S_n}({\mathbb{C}}^n)$. A central tool is the combinatorial theory of higher Specht polynomials.

Final version

References in corpus (8)

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• Representation theory and homological stability
• Hilbert schemes of 8 points
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