A Simple Bijection for the Regions of the Shi Arrangement of Hyperplanes

Abstract

The Shi arrangement ${\mathcal{S}}_n$ is the arrangement of affine hyperplanes in ${\mathbb{R}}^n$ of the form $x_i-x_j=0$ or $1$, for $1\le i<j\le n$. It dissects ${\mathbb{R}}^n$ into $(n+1{)}^{n-1}$ regions, as was first proved by Shi. We give a simple bijective proof of this result. Our bijection generalizes easily to any subarrangement of ${\mathcal{S}}_n$ containing the hyperplanes $x_i-x_j=0$ and to the extended Shi arrangements.

Cited by in corpus (1)

• Why the characteristic polynomial factors