Toric braids and (m,n)-parking functions

Abstract

We find a geometric interpretation of the ${\mathbb{A}}_{q,t}$ algebra, the algebra which appeared in the previous work of Erik Carlsson and the author on the proof of the shuffle conjecture. This allows us to construct 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 is then shown to be a corollary of this relation.