A proof of the Fields Conjectures

TL;DR

This paper proves the Fields Conjectures by analyzing the superspace coinvariant ring's structure, revealing its isomorphism to a sign-twisted permutation action, and computing its bigraded representation type.

Contribution

It establishes the isomorphism of the superspace coinvariant ring with a twisted permutation module and determines its detailed bigraded structure, confirming several conjectures.

Findings

SR_n is isomorphic to the sign-twisted permutation action of S_n.

The bigraded S_n-module structure of SR_n is explicitly computed.

The results confirm the Fields Conjectures and a related conjecture of Reiner.

Abstract

The {\em superspace ring} of rank $n$ is the algebra $\Omega_n$ of differential forms on affine $n$-space. The algebra $\Omega_n$ is bigraded with respect to polynomial and exterior degree and carries a natural action of the symmetric group $\mathfrak{S}_n$. Modding out by $\mathfrak{S}_n$-invariants with vanishing constant term yields the {\em superspace coinvariant ring} $SR_n$. We prove that, as an ungraded $\mathfrak{S}_n$-module, the space $SR_n$ is isomorphic to the sign-twisted permutation action of $\mathfrak{S}_n$ on ordered set partitions of $\{1,\dots,n\}$. We refine this result by calculating the bigraded $\mathfrak{S}_n$-isomorphism type of $SR_n$. This proves the Fields Conjectures of N. Bergeron, L. Colmenarejo, S.-X. Li, J. Machacek, R. Sulzgruber, and M. Zabrocki as well as a related conjecture of V. Reiner.