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MATHEMATICAL ENGINEERING TECHNICAL REPORTS Arrangements Stable Under the Coxeter Groups

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MATHEMATICAL ENGINEERING TECHNICAL REPORTS Arrangements Stable Under the Coxeter Groups MATHEMATICAL ENGINEERING TECHNICAL REPORTS Arrangements Stable under the Coxeter Groups Hidehiko KAMIYA, Akimichi TAKEMURA and Hiroaki TERAO METR 2011{10 March 2011 DEPARTMENT OF MATHEMATICAL INFORMATICS GRADUATE SCHOOL OF INFORMATION SCIENCE AND TECHNOLOGY THE UNIVERSITY OF TOKYO BUNKYO-KU, TOKYO 113-8656, JAPAN WWW page: http://www.keisu.t.u-tokyo.ac.jp/research/techrep/index.html The METR technical reports are published as a means to ensure timely dissemination of scholarly and technical work on a non-commercial basis. Copyright and all rights therein are maintained by the authors or by other copyright holders, notwithstanding that they have offered their works here electronically. It is understood that all persons copying this information will adhere to the terms and constraints invoked by each author's copyright. These works may not be reposted without the explicit permission of the copyright holder. Arrangements stable under the Coxeter groups Hidehiko Kamiya ∗ Faculty of Economics, Okayama University, Okayama, 700-8530, Japan Akimichi Takemura y Graduate School of Information Science and Technology, University of Tokyo, Tokyo, 113-0033, Japan Hiroaki Terao z Department of Mathematics, Hokkaido University, Sapporo, 060-0810, Japan March 2011 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[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 Kamiya, Takemura and Terao [Ranking patterns of unfolding models of codimension one, Adv. in Appl. Math. (2010)] by using our results. Keywords: all-subset arrangement, braid arrangement, characteristic polynomial, Cox- eter arrangement, Coxeter group, finite-field method, mid-hyperplane arrangement, sym- metric group. MSC2010: 20F55, 32S22, 52C35. 1 Introduction 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 Ch(B) of chambers of B. We want to find the number of W -orbits of Ch(B). A particular case of this problem was considered in ∗This work was partially supported by JSPS KAKENHI (22540134). yThis research was supported by JST CREST. zThis work was partially supported by JSPS KAKENHI (21340001). 1 the authors' previous paper (Kamiya, Takemura and Terao [9]) and the present paper is motivated by an open problem left in Section 6 of [9]. By the general results of the present paper, we give the affirmative answer to the open problem in Theorem 3.1. Suppose throughout that B\A = ;, where A = A(W ) is the Coxeter arrangement of W . In this paper, we will show that the orbit space of Ch(B) is in one-to-one correspondence with the set of chambers c of C = A[B which are contained in a, fc 2 Ch(C) j c ⊆ ag, where a 2 Ch(A) is an arbitrary chamber of A. From this fact, we find that the number of W -orbits of Ch(B) is given by jCh(C)j=jW j. On the other hand, we will also study the set of chambers c 2 Ch(C) which are contained in a chamber b 2 Ch(B) of B, fc 2 Ch(C) j c ⊆ bg. We will prove that the cardinality of this set is equal to the order of the isotropy subgroup Wb of b. Moreover, we will investigate the structure of Wb. Kamiya, Takemura and Terao [9] tried to find the number of \inequivalent ranking patterns generated by unfolding models of codimension one" in psychometrics, and ob- tained an upper bound for this number. It was left open to determine whether this upper bound is actually the exact number. The problem boils down to proving (or disprov- ing) that the orbit space of the chambers of the restricted all-subset arrangement ([9]) B under the action of the symmetric group Sm is in one-to-one correspondence with fc 2 Ch(A(Sm) [B) j c ⊆ ag for a chamber a 2 Ch(A(Sm)) of the braid arrangement A(Sm). The results of the present paper establish the one-to-one correspondence. The paper is organized as follows. In Section 2, we verify our main results. Next, in Section 3, we illustrate our general results with four examples. Those examples are mainly taken from the authors' previous studies of unfolding models in psychometrics ([5], [9]). In Section 3, we also solve the open problem of [9] (Theorem 3.1) using our general results in Section 2 applied to one of our examples. 2 Main results In this section, we state and prove our main results. Let V be a Euclidean space. Consider a Coxeter group W acting on V . Then the Coxeter arrangement A = A(W ) is the set of all reflecting hyperplanes of W . Suppose that B is a hyperplane arrangement which is stable under the natural action of W . We assume A\B = ; and define C := A[B: Let Ch(A), Ch(B) and Ch(C) denote the set of chambers of A, B and C, respectively. Define 'A : Ch(C) ! Ch(A);'B : Ch(C) ! Ch(B) by 'A(c) := the chamber of A containing c; 'B(c) := the chamber of B containing c for c 2 Ch(C). Note that the Coxeter group W naturally acts on Ch(A), Ch(B) and Ch(C). 2 Lemma 2.1. 'A and 'B are both W -equivariant, i.e., 'A(wc) = w('A(c));'B(wc) = w('B(c)) for any w 2 W and c 2 Ch(C). The proof is easy and omitted. The following result is classical (see, e.g., [3, Ch. V, x3. 2. Theorem 1 (iii) ]): Theorem 2.2. The group W acts on Ch(A) effectively and transitively. In particular, jW j = jCh(A)j. By Theorem 2.2, we can prove the following lemma. Lemma 2.3. The group W acts on Ch(C) effectively. In particular, each W -orbit of Ch(C) is of size jW j. Proof. If wc = c for w 2 W and c 2 Ch(C), then we have 'A(c) = w'A(c); which implies w = 1 by Theorem 2.2. For b 2 Ch(B), define the isotropy subgroup Wb := fw 2 W j wb = bg. Then we have −1 Lemma 2.4. For b 2 Ch(B), the group Wb acts on 'B (b) effectively and transitively. Proof. The effective part follows from Lemma 2.3, so let us prove the transitivity. −1 Let c1; c2 2 'B (b). Define A(c1; c2) := fH 2 A j c1 and c2 are on different sides of Hg: Let us prove that there exists w 2 W such that wc1 = c2 by an induction on jA(c1; c2)j. When jA(c1; c2)j = 0, we have A(c1; c2) = ; and c1 = c2. Thus we may choose w = 1. If A(c1; c2) is non-empty, then there exists H1 2 A(c1; c2) such that H1 contains a wall of c1. Let s1 denote the reflection with respect to H1. Then A(s1c1; c2) = A(c1; c2) n fH1g: By the induction assumption, there exists w1 2 W with w1s1c1 = c2. Set w := w1s1. Then wc1 = c2 and c2 = (wc1) \ c2 ⊆ (wb) \ b, which implies that (wb) \ b is not empty. Thus wb = b and w 2 Wb. The following lemma states that the W -orbits of Ch(C) and those of Ch(B) are in one-to-one correspondence. Lemma 2.5. The map 'B : Ch(C) ! Ch(B) induces a bijection from the set of W -orbits of Ch(C) to the set of W -orbits of Ch(B). Proof. For b 2 Ch(B) and c 2 Ch(C), we denote the W -orbit of b and the W -orbit of c by O(b) and by O(c), respectively. It is easy to see that 'B(O(c)) = O('B(c)); c 2 Ch(C); 3 by Lemma 2.1. Thus 'B induces a map from the set of W -orbits of Ch(C) to the set of W -orbits of Ch(B). We will show the map is bijective. Surjectivity: Let O(b) be an arbitrary orbit of Ch(B) with a representative point −1 b 2 Ch(B). Take an arbitrary c 2 'B (b). Then 'B(O(c)) = O('B(c)) = O(b); which shows the surjectivity. Injectivity: Suppose 'B(O(c1)) = 'B(O(c2)) (c1; c2 2 Ch(C)). Set bi := 'B(ci) for i = 1; 2. We have O(b1) = O('B(c1)) = 'B(O(c1)) = 'B(O(c2)) = O('B(c2)) = O(b2); so we can pick w 2 W such that wb2 = b1. Then 'B(wc2) = w('B(c2)) = wb2 = b1: −1 Therefore, both c1 and wc2 lie in 'B (b1). By Lemma 2.4, we have O(c1) = O(wc2) = O(c2): We are now in a position to state the main results of this paper.
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