A Simple Bijection For The Regions Of The Shi Arrangement Of Hyperplanes

. The Shi arrangement Sn is the arrangement of affine hyperplanes in R n of the form x i \Gamma x j = 0 or 1, for 1 i ! j n. It dissects R n into (n+1) n\Gamma1 regions, as was first proved by Shi. We give a simple bijective proof of this result. Our bijection generalizes easily to any subarrangement of Sn containing the hyperplanes x i \Gamma x j = 0 and to the extended Shi arrangements. 1. Introduction A hyperplane arrangement A is a finite set of affine hyperplanes in R n . The regions of A are the connected components of the space obtained from R n by removing the hyperplanes of A. A classical example is provided by the braid arrangement A n . It consists of the hyperplanes in R n of the form x i = x j for 1 i ! j n, i.e. the reflecting hyperplanes of the Coxeter group of type A n\Gamma1 . Its regions correspond to permutations of the set [n] := f1; 2; : : : ; ng. A deformation of A n [12] is an arrangement each of whose hyperplanes is parallel to one of the hyp..