Reflection groups acting on their hyperplanes

AbstractAfter having established elementary results on the relationship between a finite complex (pseudo-)reflection group W⊂GL(V) and its reflection arrangement A, we prove that the action of W on A is canonically related with other natural representations of W, through a ‘periodic’ family of representations of its braid group. We also prove that, when W is irreducible, then the squares of defining linear forms for A span the quadratic forms on V, which imply |A|⩾n(n+1)/2 for n=dimV, and relate the W-equivariance of the corresponding map with the period of our family