Fields conjecture on the Frobenius characteristic of the superspace coinvariant ring

From papers

Let SRnSR_n be the bigraded superspace coinvariant ring with its natural Sn\mathfrak{S}_n-action, and let grFrob(SRn;q,z)\operatorname{grFrob}(SR_n;q,z) denote its bigraded Frobenius characteristic. Let Cn,k(x;q)C_{n,k}(\mathbf{x};q) be the symmetric function defined by

Cn,k(x;q)=Δek1ent0,C_{n,k}(\mathbf{x};q)=\left.\Delta'_{e_{k-1}}e_n\right|_{t\to 0},

where Δek1\Delta'_{e_{k-1}} is the primed Delta operator. Fields conjecture on the Frobenius characteristic. The bigraded Frobenius characteristic of SRnSR_n is

grFrob(SRn;q,z)=k=1nznkCn,k(x;q).\operatorname{grFrob}(SR_n;q,z)=\sum_{k=1}^n z^{n-k}\cdot C_{n,k}(\mathbf{x};q).

This is the symmetric-function refinement of the predicted structure of SRnSR_n, relating its graded representations to Delta-operator expressions; the supplied text gives no resolution status.

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Sources & referencesView supporting material

Primary source

Satoshi Murai, Brendon Rhoades and Andy Wilson, “A proof of the Fields Conjectures”, arXiv:2505.24027 (2026).

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