The bigraded Frobenius series of the wreath product superspace coinvariant rings
Abstract
We determine the bigraded Frobenius series of the superspace coinvariant ring R(1,1)ℤrwrSn for the wreath products ℤrwrSn, which is the quotient of the ring ℂ[x1,…,xn]⊗bigwedge[θ1,…,θn] of polynomial-valued differential forms by the ideal generated by diagonal ℤrwrSn-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 Bn case, and it specializes at r=1 to recover the theorem of Murai--Rhoades--Wilson (2025) on the symmetric group Sn. 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(1,2)Sn and the determinant character of R(1,2)Bn. As a corollary, we deduce a conjecture of F. Bergeron (2024) on hook components of R(1,1)Sn.
Related
- Diagonalizable modular derivations on bigraded Poisson spaces and Koszul duality
- Finite noncommutative coinvariant algebras
- Sign components of diagonal superspace coinvariants
- Deformations of the osp(n|2)-action on the superspace of symbols of differential operators on R1|n
- Frobenius and quasi-Frobenius left Hopf algebroids
BibTeX
@misc{bhattacharya2026bigraded,
title = {{The bigraded Frobenius series of the wreath product superspace coinvariant rings}},
author = {Sutanay Bhattacharya and Sylvie Corteel and Yuhan Jiang and John Lentfer and Brendon Rhoades},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.38875},
eprint = {2610.38875},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.38875},
}
RIS
TY - UNPB
TI - The bigraded Frobenius series of the wreath product superspace coinvariant rings
AU - Sutanay Bhattacharya
AU - Sylvie Corteel
AU - Yuhan Jiang
AU - John Lentfer
AU - Brendon Rhoades
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2610.38875
UR - https://arxiv.org/abs/2610.38875
AB - We determine the bigraded Frobenius series of the superspace coinvariant ring R^(1,1)_ℤ_rwrS_n for the wreath products ℤ_rwrS_n, which is the quotient of the ring ℂ[x_1,…,x_n]⊗bigwedge[θ_1,…,θ_n] of polynomial-valued differential forms by the ideal generated by diagonal ℤ_rwrS_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 B_n case, and it specializes at r=1 to recover the theorem of Murai--Rhoades--Wilson (2025) on the symmetric group 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^(1,2)_S_n and the determinant character of R^(1,2)_B_n. As a corollary, we deduce a conjecture of F. Bergeron (2024) on hook components of R^(1,1)_S_n.
ER -