Toric braids and $(m,n)$-parking functions

TL;DR

This paper interprets the Dyck path algebra as a representation of toric braids and connects sums over $(m,n)$-parking functions to braid evaluations, deriving the compositional shuffle conjecture as a corollary.

Contribution

It provides a novel interpretation of the Dyck path algebra in terms of toric braid representations and links parking functions to braid evaluations, proving the shuffle conjecture as a consequence.

Findings

Dyck path algebra viewed as a toric braid representation

Sums over $(m,n)$-parking functions relate to braid evaluations

Proves the compositional $(km,kn)$-shuffle conjecture as a corollary

Abstract

The Dyck path algebra construction of Carlsson and Mellit from arXiv:1508.06239 is interpreted as a representation of "the positive part" of the group of toric braids. Then certain sums over $(m,n)$-parking functions are related to evaluations of this representation on some special braids. The compositional $(km,kn)$-shuffle conjecture of Bergeron, Garsia, Leven and Xin from arXiv:1404.4616 is then shown to be a corollary of this relation.