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A Geometric Realization of Partially-Symmetric Macdonald Polynomials
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abstract
We formulate a precise conjecture relating integral form partially-symmetric Macdonald polynomials and the parabolic flag Hilbert schemes of Carlsson, Gorsky, and Mellit. This extends, in an explicit fashion, Haiman's realization of modified Macdonald symmetric functions via Hilbert schemes of points in the plane. As evidence for our conjecture we prove that it is compatible with the action of certain elements in Carlsson and Mellit's algebra $\mathbb{A}_{t,q}$, including degree $1$ Pieri formulas.
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Flagged LLT polynomials, nonsymmetric plethysm, and nonsymmetric Macdonald polynomials
A new class of flagged LLT polynomials and a nonsymmetric plethysm operator yield a conjectural nonsymmetric strengthening of Macdonald positivity.
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