The bigraded Frobenius series of the wreath product superspace coinvariant rings

Abstract

We determine the bigraded Frobenius series of the superspace coinvariant ring $R_{{\mathbb{Z}}_r\wr {\mathfrak{S}}_n}^{(1,1)}$ for the wreath products ${\mathbb{Z}}_r\wr {\mathfrak{S}}_n$, which is the quotient of the ring $\mathbb{C}[x_1,\dots ,x_n]\otimes \bigwedge [{\theta }_1,\dots ,{\theta }_n]$ of polynomial-valued differential forms by the ideal generated by diagonal ${\mathbb{Z}}_r\wr {\mathfrak{S}}_n$-invariants with vanishing constant term. The bigraded multiplicity of each irreducible character is a sum over standard Young $r$-tableaux of an expression determined by the flag major index and flag descent number. This refines the Frobenius series graded by fermionic degree found by Bhattacharya--Rhoades (2026). It specializes at $r=2$ to the hyperoctahedral group ${\mathfrak{B}}_n$ case, and it specializes at $r=1$ to recover the theorem of Murai--Rhoades--Wilson (2025) on the symmetric group ${\mathfrak{S}}_n$. We give new formulas for the bigraded multiplicities of hook characters of the superspace coinvariant ring in types $A$ and $B$, yielding upper bounds on trigraded multiplicities of the sign character of $R_{{\mathfrak{S}}_n}^{(1,2)}$ and the determinant character of $R_{{\mathfrak{B}}_n}^{(1,2)}$. As a corollary, we deduce a conjecture of F. Bergeron (2024) on hook components of $R_{{\mathfrak{S}}_n}^{(1,1)}$.

arXiv comment43 pages, 2 figures