A proof of the fermionic Theta coinvariant conjecture

2022/02/08 by Alessandro Iraci, Iraci, Alessandro, Brendon Rhoades +3
Mathematics · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2202.04170

openalex publication_date 2022/02/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let (x₁, …, x_n, y₁, …, y_n) be a list of 2n commuting variables, (θ₁, …, θ_n, ξ₁, …, ξ_n) be a list of 2n anticommuting variables, and ℂ[X_n, Y_n] ⊗ \wedge \Θ_n, Ξ_n\ be the algebra generated by these variables. D'Adderio, Iraci, and Vanden Wyngaerd introduced the \em Theta operators on the ring of symmetric functions and used them to conjecture a formula for the quadruply-graded \mathfrakS_n-isomorphism type of ℂ[X_n,Y_n] ⊗ \wedge \Θ_n, Ξ_n\/I where I is the ideal generated by \mathfrakS_n-invariants with vanishing constant term. We prove their conjecture in the `purely fermionic setting' obtained by setting the commuting variables equal x_i, y_i equal to zero.

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