John Lentfer, UC Berkeley

Title: The (1,2)-bosonic-fermionic coinvariant ring

Abstract: In 1994 Haiman introduced the ring of diagonal coinvariants, which is a quotient of a polynomial ring in two sets of commutative variables by invariants of the diagonal action of the symmetric group.Recently, there has been much interest in studying a more general class of coinvariant rings with k sets of n commutative (bosonic) variables and j sets of n anticommutative (fermionic) variables; denote this ring by $R_n^{(k,j)}$.
We will focus on the coinvariant ring $R_n^{(1,2)}$, with one set of bosonic and two sets of fermionic variables.
By interpolating 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 propose a monomial basis for $R_n^{(1,2)}$.
We use the proposed basis to give combinatorial formulas for its conjectural Hilbert series and Frobenius series.
We will explain how our work on $R_n^{(1,2)}$ relates to the Theta conjecture and recent work of Iraci, Nadeau, and Vanden Wyngaerd (2023).