A proof of the fermionic Theta coinvariant conjecture

TL;DR

This paper proves a conjecture related to fermionic Theta coinvariants, specifically in the purely fermionic setting where commuting variables are set to zero, advancing understanding in algebraic combinatorics.

Contribution

The paper provides a proof of the Theta coinvariant conjecture in the purely fermionic case, confirming a key formula involving symmetric functions and anticommuting variables.

Findings

Confirmed the Theta coinvariant conjecture in the fermionic setting.

Established the structure of fermionic Theta coinvariants.

Validated the conjecture's formula for the algebra with anticommuting variables.

Abstract

Let $(x_1, \dots, x_n, y_1, \dots, y_n)$ be a list of $2n$ commuting variables, $(\theta_1, \dots, \theta_n, \xi_1, \dots, \xi_n)$ be a list of $2n$ anticommuting variables, and $\mathbb{C}[X_n, Y_n] \otimes \wedge \{\Theta_n, \Xi_n\}$ 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 $\mathfrak{S}_n$-isomorphism type of $\mathbb{C}[X_n,Y_n] \otimes \wedge \{\Theta_n, \Xi_n\}/I$ where $I$ is the ideal generated by $\mathfrak{S}_n$-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.