Arrangements stable under the Coxeter groups

TL;DR

This paper investigates the structure of chambers in arrangements stable under Coxeter groups, establishing a correspondence between W-orbits and chambers, and solving an open problem related to ranking patterns.

Contribution

It introduces a new correspondence between chambers of arrangements stable under Coxeter groups and those of their unions, providing formulas for counting chambers and orbits, and addresses an open problem in the field.

Findings

W-orbits of chambers correspond to chambers of a union arrangement within a fixed chamber.

Number of W-orbits equals the total chambers divided by the order of W.

Cardinality of chambers contained in a given chamber equals the isotropy subgroup order.

Abstract

Let B be a real hyperplane arrangement which is stable under the action of a Coxeter group W. Then B 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\cup 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 W_b of b. We illustrate these results with some examples, and solve an open problem in Kamiya, Takemura and Terao [Ranking patterns of unfolding models of codimension one, Adv. in Appl. Math. (2010)] by using our results.