A simple bijection for the regions of the Shi arrangement of hyperplanes Original Research Article

The Shi arrangement Ln is the arrangement of affine hyperplanes in Rn of the form xi − xj = 0 or 1, for 1 ⩽ i < j ⩽ n. It dissects Rn 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 Ln containing the hyperplanes xi − xj = 0 and to the extended Shi arrangements. It also implies the fact that the number of regions of Ln which are relatively bounded is (n − 1)n−1.