Toric braids and $(m,n)$-parking functions
Combinatorics
2022-08-10 v1 Quantum Algebra
Representation Theory
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 -parking functions are related to evaluations of this representation on some special braids. The compositional -shuffle conjecture of Bergeron, Garsia, Leven and Xin from arXiv:1404.4616 is then shown to be a corollary of this relation.
Cite
@article{arxiv.1604.07456,
title = {Toric braids and $(m,n)$-parking functions},
author = {Anton Mellit},
journal= {arXiv preprint arXiv:1604.07456},
year = {2022}
}