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

Lentfer, John

doi:10.5802/alco.424

Lentfer, John ¹

¹ Department of Mathematics University of California, Berkeley Berkeley, CA (USA)

Algebraic Combinatorics, Volume 8 (2025) no. 3, pp. 711-743

Abstract

We give the first conjectural construction of a monomial basis for the coinvariant ring $R_n^{(1,2)}$, for the symmetric group $\mathfrak{S}_n$ 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 $4^nn!$.

Received: 2024-08-09
Revised: 2025-02-19
Accepted: 2025-02-24
Published online: 2025-06-26

DOI: 10.5802/alco.424

Classification: 01A35, 14A15, 11L05
Keywords: Coinvariant rings, Artin basis, Frobenius series, Hilbert series, symmetric functions

Author's affiliations:

Lentfer, John ¹

¹ Department of Mathematics University of California, Berkeley Berkeley, CA (USA)

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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Lentfer, John. A conjectural basis for the $(1,2)$-bosonic-fermionic coinvariant ring. Algebraic Combinatorics, Volume 8 (2025) no. 3, pp. 711-743. doi: 10.5802/alco.424

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