A Geometric Realization of Partially-Symmetric Macdonald Polynomials

2023/12/18 by Ben Goodberry, Goodberry, Ben, Daniel L. Orr +1 · 1 citation
Mathematics · #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Advanced Mathematical Identities

paper · pdf · doi:10.48550/arxiv.2312.11657

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 \mathbbA_t,q, including degree 1 Pieri formulas.

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