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arXiv · 2503.19323

Invariant theory for wreath products acting on superpolynomials

Trevor Karn·Victor Reiner
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Abstract

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.

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Trevor Karn, Victor Reiner. 2025-03-25. Invariant theory for wreath products acting on superpolynomials. https://arxiv.org/abs/2503.19323

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