Zabrocki's superspace diagonal coinvariant conjecture

Let Ωn[y1,,yn]=C[x1,,xn]C[y1,,yn]{θ1,,θn}\Omega_n[y_1,\dots,y_n]=\mathbb{C}[x_1,\dots,x_n]\otimes\mathbb{C}[y_1,\dots,y_n]\otimes\bigwedge\{\theta_1,\dots,\theta_n\}, with SnS_n acting triply diagonally, and let

SDRn=Ωn[y1,,yn]/Ωn[y1,,yn]+SnSDR_n=\Omega_n[y_1,\dots,y_n]/\langle\Omega_n[y_1,\dots,y_n]^{S_n}_+\rangle

be the superspace diagonal coinvariant algebra. Let grFrob(;q,t,z)\operatorname{grFrob}(-;q,t,z) denote the triply graded Frobenius image, where q,t,zq,t,z track xx-, yy-, and θ\theta-degrees. Zabrocki's conjecture.

grFrob(SDRn;q,t,z)=k=1nznkΔek1en.\operatorname{grFrob}(SDR_n;q,t,z)=\sum_{k=1}^n z^{n-k}\Delta'_{e_{k-1}}e_n.

This conjecture proposes SDRnSDR_n as an algebraic model for the full, rather than t=0t=0 specialized, Delta-operator expression. The source does not specify a resolution status.

Sources & referencesView supporting material

Primary source

Brendon Rhoades, “Generalizations of the flag variety tied to the Macdonald-theoretic delta operators”, arXiv:2204.03386 (2024).

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