Rhoades–Wilson double superspace Vandermonde conjecture

For 1kn1\leq k\leq n, let Vn,k\mathbb{V}_{n,k} be the smallest subspace of Ωn[y1,,yn]\Omega_n[y_1,\dots,y_n] containing the superspace Vandermonde δn,k\delta_{n,k}, closed under /xi\partial/\partial x_i and /yi\partial/\partial y_i, and closed under each polarization operator

y1jx1j++ynjxnj,j1.y_1\frac{\partial^j}{\partial x_1^j}+\cdots+y_n\frac{\partial^j}{\partial x_n^j},\qquad j\geq1.

Let SDRnSDR_n be the superspace diagonal coinvariant algebra. Rhoades–Wilson double Vandermonde conjecture.

grFrob(Vn,k;q,t)=Δek1en,\operatorname{grFrob}(\mathbb{V}_{n,k};q,t)=\Delta'_{e_{k-1}}e_n,

and the composite map

k=1nVn,kΩn[y1,,yn]SDRn\bigoplus_{k=1}^n\mathbb{V}_{n,k}\hookrightarrow\Omega_n[y_1,\dots,y_n]\twoheadrightarrow SDR_n

is bijective. The source notes that the first assertion is known at t=0t=0, while the corresponding coinvariant statement remains open.

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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