A SIMPLE BIJECTION FOR THE REGIONS OF THE SHI ARRANGEMENT OF HYPERPLANES

CHRISTOS A. ATHANASIADIS SVANTE LINUSSON

Abstract. The Shi arrangement S_{n} is the arrangement of affine hyperplanes in ℝ^{n} of the form x_{i}−x_{j} = 0 or 1, for 1 ≤ i < j ≤ n. It dissects 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 S_{n} containing the hyperplanes x_{i} − x_{j} = 0 and to the extended Shi arrangements.