Generalized Pak-Stanley labeling for multigraphical hyperplane arrangements for n = 3

Abstract

Pak-Stanley labeling was originally defined by Pak and Stanley in 1998 as a bijective map from the set of regions of an extended Shi arrangement to the set of parking functions. Later this map was generalized to other hyperplane arrangements associated with graphs and directed multigraphs, but this map is not necessarily bijective in these more general cases. It was shown by in Sam Hopkins and David Perkinson in 2016 and Mikhail Mazin in 2017 that Pak-Stanley labeling is surjective to the set of G-parking functions, where G is the directed multigraph associated with the hyperplane arrangement. This leads to the natural question of when the generalized Pak-Stanley map is bijective. We determine a necessary condition for a directed multigraph to admit a hyperplane arrangement that admits a injective Pak-Stanley labeling. For the special case n = 3, we present examples of directed multigraphs that satisfy our necessary condition but only admit hyperplane arrangements with a non-injective Pak-Stanley labeling, showing that the condition is not sufficient.