The $\mathbb{A}_{q,t}$ algebra and parabolic flag Hilbert schemes

TL;DR

This paper constructs an action of the algebra $\\mathbb{A}_{q,t}$ on the equivariant K-theory of specific strata in flag Hilbert schemes, linking algebraic structures with geometric and combinatorial data.

Contribution

It introduces a new geometric realization of the algebra $\mathbb{A}_{q,t}$ acting on the K-theory of flag Hilbert schemes, connecting algebraic, geometric, and combinatorial frameworks.

Findings

Fixed points correspond to generalized Macdonald polynomials

Explicit combinatorial formulas for matrix elements of operators

New geometric interpretation of the algebra $\mathbb{A}_{q,t}$

Abstract

The earlier work of the first and the third named authors introduced the algebra $\mathbb{A}_{q,t}$ and its polynomial representation. In this paper we construct an action of this algebra on the equivariant K-theory of certain smooth strata in the flag Hilbert schemes of points on the plane. In this presentation, the fixed points of torus action correspond to generalized Macdonald polynomials and the the matrix elements of the operators have explicit combinatorial presentation.