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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2505.24027 |
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- The {\em superspace ring} of rank $n$ is the algebra $Ω_n$ of differential forms on affine $n$-space. The algebra $Ω_n$ 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.