Hyperplane arrangements between Shi and Ish

Rui Duarte

doi:10.1016/j.endm.2018.06.046

Abstract

We introduce a new family of hyperplane arrangements in dimension n≥3 that includes both the Shi arrangement and the Ish arrangement. We prove that all the members of this family have the same number of regions — the connected components of the complement of the union of the hyperplanes — which can be bijectively labeled with the Pak-Stanley labelling. In addition, we characterise the Pak-Stanley labels of the regions of this family of hyperplane arrangements.

Highlights

In this paper we introduce a family of arrangements of hyperplanes in general dimension “between Ish and Shi”, that is, formed by hyperplanes that are hyperplanes of the Shi arrangement or hyperplanes of the Ish arrangement, all of the same dimension, and characterise the Pak-Stanley labels of the regions of these arrangements of hyperplanes
Note that Cij | 1 ≤ i < j ≤ n = Coxn is the n-dimensional Coxeter arrangement, A2n = Shin is the n-dimensional Shi arrangement, and Ann = Ishn is the n-dimensional Ish arrangement introduced by Armstrong [2]
We represent both A23 = Shi3 and A33 = Ish3 on Figure 1. We study these arrangements with special interest in the Pak-Stanley labelings of the regions of the arrangements

Summary

Introduction

In this paper we introduce a family of arrangements of hyperplanes in general dimension “between Ish and Shi”, that is, formed by hyperplanes that are hyperplanes of the Shi arrangement or hyperplanes of the Ish arrangement, all of the same dimension, and characterise the Pak-Stanley labels of the regions of these arrangements of hyperplanes. Xn) ∈ Rn | x1 = xj + i and define, for 2 ≤ k ≤ n, Akn := Cij | 1 ≤ i < j ≤ n ∪ Iij | 1 ≤ i < j ≤ n ∧ i < k ∪ Sij | k ≤ i < j ≤ n. We represent both A23 = Shi and A33 = Ish on Figure 1 We study these arrangements with special interest in the Pak-Stanley labelings of the regions of the arrangements. The labels are G-parking functions for special directed multi-graphs G as defined by Mazin [8]. Hopkins and Perkinson [6] showed that the labels of the Pak-Stanley labeling of the regions of a given hyperplane arrangement defined by G are exactly the G-parking functions, a fact that had been conjectured by Duval, Klivans and Martin [5]. Whereas Hopkins and Perkinson’s hyperplane arrangements include for example the (original) multidimensional Shi arrangement, Mazin’s hyperplane arrangements include the multidimensional k-Shi arrangement, the multidimensional Ish arrangement and all the arrangements we consider here

The characteristic polynomial

The Pak-Stanley labelling

Partial Parking Functions