2024/10/17 by Daniel L. Orr, Orr, Daniel, Milo Bechtloff Weising +1 · 1 citation
Mathematics · Physics and Astronomy · #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Nonlinear Waves and Solitons #Quantum Algebra (math.QA) #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2410.13642
openalex publication_date 2024/10/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The modified Macdonald functions \widetildeHμ are fundamental objects in modern algebraic combinatorics. Haiman showed that there is a correspondence between the (ℂ*)2-fixed points Iμ of the Hilbert schemes Hilbn(ℂ2) and the functions \widetildeHμ realizing a derived equivalence between (ℂ*)2-equivariant coherent sheaves on Hilbn(ℂ2) and (\mathfrakSn × (ℂ*)2)-equivariant coherent sheaves on (ℂ2)n. Carlsson--Gorsky--Mellit introduced a larger family of smooth projective varieties PFHn,n-k called the parabolic flag Hilbert schemes. They showed that an algebra \mathbbBq,t, directly related to the double Dyck path algebra \mathbbAq,t employed in Carlsson--Mellit's proof of the Shuffle Theorem, acts naturally on the (ℂ*)2-equivariant K-theory U\bullet of these spaces and, moreover, there is a \mathbbBq,t-isomorphism Φ: U\bullet → V\bullet where V\bullet is the polynomial representation. The isomorphism Φ: U\bullet → V\bullet is known to extend Haiman's correspondence. In this paper, we explicitly compute the images Φ(Hμ,w) of the normalized (ℂ*)2-fixed point classes Hμ,w of the spaces PFHn,n-k and show they agree with the modified partially symmetric Macdonald polynomials \widetildeH(λ|γ) introduced by Goodberry-Orr, confirming their prior conjecture. We use this result to give an explicit formula for the action of the involution N on V\bullet.