Symmetric Ideals and Invariant Hilbert Schemes

TL;DR

This paper studies symmetric ideals and invariant Hilbert schemes, revealing their geometric properties, classifying ideals, and analyzing singularities using combinatorial tools, advancing understanding of symmetric polynomial invariants.

Contribution

It provides a comprehensive geometric analysis of zero-dimensional symmetric ideals and invariant Hilbert schemes, including irreducibility, smoothness, and classification results for specific modules.

Findings

Invariant Hilbert schemes are irreducible or smooth for certain modules.

Classified all homogeneous symmetric ideals within a range.

Identified singular points of the 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 $\dim \rho \leq 2n$ 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.