A proof of the Fields Conjectures

Abstract

The {\em superspace ring} of rank $n$ is the algebra $\hat{I}{\text{c◯}}_n$ of differential forms on affine $n$-space. The algebra $\hat{I}{\text{c◯}}_n$ 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} $S{R}_n$. We prove that, as an ungraded ${\mathfrak{S}}_n$-module, the space $S{R}_n$ 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 $S{R}_n$. 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.

28 pages. Version 2 corrects several typos in Version 1