Theoretical analysis connects partially-symmetric Macdonald polynomials with parabolic flag Hilbert schemes, extending geometric frameworks for algebraic combinatorics.
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 \(At,q\), including degree \(1\) Pieri formulas. Mathematics Subject Classifications: 05E10, 05E05 Keywords: Macdonald polynomials, Hilbert schemes, Hecke algebras
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Goodberry et al. (2026) studied this question.
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