Wreath Macdonald polynomials, a survey

TL;DR

This survey reviews the theory of wreath Macdonald polynomials, their geometric origins, recent algebraic tools for their study, and introduces new conjectures, highlighting progress and open questions in the field.

Contribution

It provides a comprehensive overview of wreath Macdonald polynomials, including recent advances using quantum toroidal algebra representation theory and new conjectures on their properties.

Findings

Wreath Macdonald polynomials generalize classical Macdonald functions via geometric and algebraic methods.

Recent tools from quantum toroidal algebra facilitate explicit access to wreath Macdonald polynomials.

The paper formulates new conjectures on constants in wreath Macdonald P-polynomials.

Abstract

Wreath Macdonald polynomials arise from the geometry of $\Gamma$-fixed loci of Hilbert schemes of points in the plane, where $\Gamma$ is a finite cyclic group of order $r\ge 1$. For $r=1$, they recover the classical (modified) Macdonald symmetric functions through Haiman's geometric realization of these functions. The existence, integrality, and positivity of wreath Macdonald polynomials for $r>1$ was conjectured by Haiman and first proved in work of Bezrukavnikov and Finkelberg by means of an equivalence of derived categories. Despite the power of this approach, a lack of explicit tools providing direct access to wreath Macdonald polynomials -- in the spirit of Macdonald's original works -- has limited progress in the subject. A recent result of Wen provides a remarkable set of such tools, packaged in the representation theory of quantum toroidal algebras. In this article, we survey Wen's result along with the basic theory of wreath Macdonald polynomials, including its geometric foundations and the role of bigraded reflection functors in the construction of wreath analogs of the $\nabla$ operator. We also formulate new conjectures on the values of important constants arising in the theory of wreath Macdonald $P$-polynomials. A variety of examples are used to illustrate these objects and constructions throughout the paper.