A proof of the Fields Conjectures

2025/05/29 by Murai, Satoshi, Rhoades, Brendon, Wilson, Andy
#Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2505.24027

Abstract

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 \mathfrakS_n. Modding out by \mathfrakS_n-invariants with vanishing constant term yields the \em superspace coinvariant ring SR_n. We prove that, as an ungraded \mathfrakS_n-module, the space SR_n is isomorphic to the sign-twisted permutation action of \mathfrakS_n on ordered set partitions of \1,…,n\. We refine this result by calculating the bigraded \mathfrakS_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.

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Related

• Superspace coinvariants and hyperplane arrangements
• The superspace coinvariant ring of type B
• Vandermondes in superspace
• Set superpartitions and superspace duality modules
• On automorphisms of affine superspaces