Invariant theory for wreath products acting on superpolynomials
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866908403308167168 |
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| author | Karn, Trevor Reiner, Victor |
| author_facet | Karn, Trevor Reiner, Victor |
| contents | This paper considers a finite group $G$ acting linearly on the variables $V$ of a polynomial algebra, or an exterior algebra, or superpolynomial algebra with both commuting and anticommuting variables. In this setting, the Hilbert series for the $G$-invariant subalgebra turns out to determine the analogous Hilbert series for the wreath product $P[G]$ acting on $V^n$ for any permutation group $P$ inside the symmetric group $S_n$ on $n$ letters.
This leads to a structural result: one can collate the direct sum for all $n$ of the $S_n[G]$-invariant subalgebras to form a graded ring via an external shuffle product, whose structure turns out to be a superpolynomial algebra generated by the $G$-invariants. A parallel statement holds for the direct sum of all $S_n[G]$-antiinvariants, which forms a graded ring via an external signed shuffle product, isomorphic to the superexterior algebra generated by the $G$-invariants. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_19323 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Invariant theory for wreath products acting on superpolynomials Karn, Trevor Reiner, Victor Combinatorics Commutative Algebra 05E40, 13A50, 16W22 This paper considers a finite group $G$ acting linearly on the variables $V$ of a polynomial algebra, or an exterior algebra, or superpolynomial algebra with both commuting and anticommuting variables. In this setting, the Hilbert series for the $G$-invariant subalgebra turns out to determine the analogous Hilbert series for the wreath product $P[G]$ acting on $V^n$ for any permutation group $P$ inside the symmetric group $S_n$ on $n$ letters. This leads to a structural result: one can collate the direct sum for all $n$ of the $S_n[G]$-invariant subalgebras to form a graded ring via an external shuffle product, whose structure turns out to be a superpolynomial algebra generated by the $G$-invariants. A parallel statement holds for the direct sum of all $S_n[G]$-antiinvariants, which forms a graded ring via an external signed shuffle product, isomorphic to the superexterior algebra generated by the $G$-invariants. |
| title | Invariant theory for wreath products acting on superpolynomials |
| topic | Combinatorics Commutative Algebra 05E40, 13A50, 16W22 |
| url | https://arxiv.org/abs/2503.19323 |