Séminaire Lotharingien de Combinatoire, 93B.129 (2025), 12 pp.

John Lentfer

A Conjectural Basis for the (1,2)-Bosonic-Fermionic Coinvariant Ring

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. (2024). We prove that our proposed basis has cardinality 2^n-1n!, 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), including formulas for hook characters and the m_μ coefficients.

Received: November 15, 2024. Accepted: February 15, 2025. Final version: April 1, 2025.

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