A proof of the Square Paths Conjecture

TL;DR

This paper proves the Square Paths Conjecture by extending combinatorial frameworks related to modified Macdonald polynomials and the $ abla$ operator, connecting algebraic and combinatorial properties of symmetric functions.

Contribution

It introduces an extension of schedules to labeled square paths and uses this to prove the Square Paths Conjecture, advancing understanding of Macdonald polynomial combinatorics.

Findings

Proved the Square Paths Conjecture.

Extended the notion of schedules to labeled square paths.

Connected algebraic properties of symmetric functions to combinatorial structures.

Abstract

The modified Macdonald polynomials, introduced by Garsia and Haiman (1996), have many astounding combinatorial properties. One such class of properties involves applying the related $\nabla$ operator of Bergeron and Garsia (1999) to basic symmetric functions. The first discovery of this type was the (recently proven) Shuffle Conjecture of Haglund, Haiman, Loehr, Remmel, and Ulyanov (2005), which relates the expression $\nabla e_n$ to parking functions. In (2007), Loehr and Warrington conjectured a similar expression for $\nabla p_n$ in terms of labeled square paths. In this paper, we extend Haglund and Loehr's (2005) notion of schedules to labeled square paths and apply this extension to prove the Square Paths Conjecture.