Fields Group superspace harmonic-generation conjecture

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Let dj:Ωn→Ωnd_j:\Omega_n\to\Omega_n be defined for j≥1j\geq1 by

dj(f)=∑i=1nθi∂jf∂xij,d_j(f)=\sum_{i=1}^n\theta_i\frac{\partial^j f}{\partial x_i^j},

and let δn∈C[x1,…,xn]\delta_n\in\mathbb{C}[x_1,\dots,x_n] be the Vandermonde determinant. Let SRnSR_n be the superspace coinvariant algebra and identify its harmonic space with the orthogonal complement of its defining ideal. Fields Group harmonic-generation conjecture. The harmonic space of SRnSR_n is generated by the 2n−12^{n-1} elements

d1ϵ1d2ϵ2⋯dn−1ϵn−1(δn),d_1^{\epsilon_1}d_2^{\epsilon_2}\cdots d_{n-1}^{\epsilon_{n-1}}(\delta_n),

where each ϵi∈{0,1}\epsilon_i\in\{0,1\}. These elements are known to be harmonic, but the claimed generation and the resulting structure remain unresolved in the source.

References

Primary source

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

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