A proof of the Fields Conjectures
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866912403708444672 |
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| author | Murai, Satoshi Rhoades, Brendon Wilson, Andy |
| author_facet | Murai, Satoshi Rhoades, Brendon Wilson, Andy |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_24027 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A proof of the Fields Conjectures Murai, Satoshi Rhoades, Brendon Wilson, Andy Combinatorics Representation Theory 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. |
| title | A proof of the Fields Conjectures |
| topic | Combinatorics Representation Theory |
| url | https://arxiv.org/abs/2505.24027 |