A geometric realization of partially-symmetric Macdonald polynomials
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. Mathematics Subject Classifications: 05E10, 05E05 Keywords: Macdonald polynomials, Hilbert schemes, Hecke algebras
Fields of Study & Subjects
Advanced Combinatorial Mathematics
Algebraic structures and combinatorial models
Advanced Mathematical Identities
Publication details
- Journal / source
- Combinatorial Theory
- Volume
- 6
- Issue
- 2
- Publication date
- 20 August 2026
- Language
- EN