A conjectural basis for the $(1,2)$-bosonic-fermionic coinvariant ring

TL;DR

This paper proposes a conjectural monomial basis for the coinvariant ring with mixed bosonic and fermionic variables, connecting previous bases and providing combinatorial and representation-theoretic insights.

Contribution

It introduces the first conjectural basis for the $(1,2)$-bosonic-fermionic coinvariant ring, linking it to existing bases and conjectures, and extends results to type $B_n$.

Findings

Basis has cardinality $2^{n-1}n!$, matching a conjecture on the dimension.

Provides a combinatorial expression for the Hilbert series.

Establishes a bijection linking the basis to segmented permutations, supporting conjectured Frobenius series.

Abstract

We give the first conjectural construction of a monomial basis for the coinvariant ring $R_n^{(1,2)}$, for the symmetric group $S_n$ acting on one set of bosonic (commuting) and two sets of fermionic (anticommuting) variables. Our construction interpolates between the modified Motzkin path basis for $R_n^{(0,2)}$ of Kim-Rhoades (2022) and the super-Artin basis for $R_n^{(1,1)}$ conjectured by Sagan-Swanson (2024) and proven by Angarone et al. (2025). We prove that our proposed basis has cardinality $2^{n-1}n!$, aligning with a conjecture of Zabrocki (2020) on the dimension of $R_n^{(1,2)}$, and show how it gives a combinatorial expression for the Hilbert series. We also conjecture a Frobenius series for $R_n^{(1,2)}$. We show that these proposed Hilbert and Frobenius series are equivalent to conjectures of Iraci, Nadeau, and Vanden Wyngaerd (2024) on $R_n^{(1,2)}$ in terms of segmented Smirnov words, by exhibiting a weight-preserving bijection between our proposed basis and their segmented permutations. We extend some of their results on the sign character to hook characters, and give a formula for the $m_\mu$ coefficients of the conjectural Frobenius series. Finally, we conjecture a monomial basis for the analogous ring in type $B_n$, and show that it has cardinality $4^nn!$.