Invariant theory for wreath products acting on superpolynomials

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.