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

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

1997-02-20

arxiv.org/abs/math/9702224v1

Mathematics

Combinatorics

Scientific paper

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 $1 \leq i < j \leq n$. 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.

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