Arrangements stable under the coxeter groups

Abstract

Let B be a real hyperplane arrangement which is stable under the action of a Coxeter group W. Then W acts naturally on the set of chambers of B. We assume that B is disjoint from the Coxeter arrangement A = A(W) of W. In this paper, we show that the W-orbits of the set of chambers of B are in one-to-one correspondence with the chambers of C = A {n-ary union} B which are contained in an arbitrarily fixed chamber of A. From this fact, we find that the number of W-orbits of the set of chambers of B is given by the number of chambers of C divided by the order of W. We will also study the set of chambers of C which are contained in a chamber b of B. We prove that the cardinality of this set is equal to the order of the isotropy subgroup Wb of b. We illustrate these results with some examples, and solve an open problem in [H. Kamiya, A. Takemura, H. Terao, Ranking patterns of unfolding models of codimension one, Adv. in Appl. Math. 47 (2011) 379-400] by using our results.