Multigraph Hyperplane Arrangements and Parking Functions

TL;DR

This paper generalizes the Pak-Stanley labeling of hyperplane arrangement regions to oriented multigraphs, providing a simple proof of bijectivity for arrangements associated with any k, extending previous results.

Contribution

It introduces a new generalization of the G-Shi arrangement to oriented multigraphs, proving the bijectivity of the labeling in this broader context.

Findings

Generalized Pak-Stanley labeling to oriented multigraphs.

Proved bijectivity of the labeling for arrangements with arbitrary k.

Provided a straightforward proof extending previous bijectivity results.

Abstract

Back in the nineties Pak and Stanley introduced a labeling of the regions of a k-Shi arrangement by k-parking functions and proved its bijectivity. Duval, Klivans, and Martin considered a modification of this construction associated with a graph G. They introduced the G-Shi arrangement and a labeling of its regions by G-parking functions. They conjectured that their labeling is surjective, i.e. that every G-parking function appears as a label of a region of the G-Shi arrangement. Later Hopkins and Perkinson proved this conjecture. In particular, this provided a new proof of the bijectivity of Pak-Stanley labeling in the k=1 case. We generalize Hopkins-Perkinson's construction to the case of arrangements associated with oriented multigraphs. In particular, our construction provides a simple straightforward proof of the bijectivity of the original Pak-Stanley labeling for arbitrary k.