We find a geometric interpretation of the 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 -parking functions are related to evaluations of this representation on some special braids. The compositional -shuffle conjecture of Bergeron, Garsia, Leven, and Xin is then shown to be a corollary of this relation.
"Toric braids and -parking functions." Duke Math. J. 170 (18) 4123 - 4169, 1 December 2021. https://doi.org/10.1215/00127094-2021-0011