Symmetric Ideals and Invariant Hilbert Schemes
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2024
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| _version_ | 1866908533679718400 |
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| author | Debus, Sebastian Kretschmer, Andreas |
| author_facet | Debus, Sebastian Kretschmer, Andreas |
| contents | 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 $\mathrm{Hilb}_ρ^{S_n}(\mathbb{C}^n)$ parametrizing symmetric subschemes of $\mathbb{C}^n$ whose coordinate rings, as $S_n$-modules, are isomorphic to a given representation $ρ$. In the case that $ρ= M^λ$ is a permutation module corresponding to certain special types of partitions $λ$ of $n$, we prove that $\mathrm{Hilb}_ρ^{S_n}(\mathbb{C}^n)$ is irreducible or even smooth. We also prove irreducibility whenever $\dim ρ\leq 2n$ 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 $\mathrm{Hilb}_ρ^{S_n}(\mathbb{C}^n)$. A central tool is the combinatorial theory of higher Specht polynomials. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_15240 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Symmetric Ideals and Invariant Hilbert Schemes Debus, Sebastian Kretschmer, Andreas Algebraic Geometry Commutative Algebra Combinatorics 14C05, 14L30, 05E14, 05E40 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 $\mathrm{Hilb}_ρ^{S_n}(\mathbb{C}^n)$ parametrizing symmetric subschemes of $\mathbb{C}^n$ whose coordinate rings, as $S_n$-modules, are isomorphic to a given representation $ρ$. In the case that $ρ= M^λ$ is a permutation module corresponding to certain special types of partitions $λ$ of $n$, we prove that $\mathrm{Hilb}_ρ^{S_n}(\mathbb{C}^n)$ is irreducible or even smooth. We also prove irreducibility whenever $\dim ρ\leq 2n$ 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 $\mathrm{Hilb}_ρ^{S_n}(\mathbb{C}^n)$. A central tool is the combinatorial theory of higher Specht polynomials. |
| title | Symmetric Ideals and Invariant Hilbert Schemes |
| topic | Algebraic Geometry Commutative Algebra Combinatorics 14C05, 14L30, 05E14, 05E40 |
| url | https://arxiv.org/abs/2404.15240 |