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Maximal Subgroups of the Coxeter Group W(H4) and Quaternions

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Maximal Subgroups of the Coxeter Group W(H4) and Quaternions View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Linear Algebra and its Applications 412 (2006) 441–452 www.elsevier.com/locate/laa Maximal subgroups of the Coxeter group W(H4) and quaternions Mehmet Koca a, Ramazan Koç b,∗, Muataz Al-Barwani a, Shadia Al-Farsi a aDepartment of Physics, College of Science, Sultan Qaboos University, P.O. Box 36, Al-Khod 123, Muscat, Oman bDepartment of Physics, Faculty of Engineering, Gaziantep University, 27310 Gaziantep, Turkey Received 9 October 2004; accepted 17 July 2005 Available online 12 September 2005 Submitted by H. Schneider Abstract The largest finite subgroup of O(4) is the non-crystallographic Coxeter group W(H4) of order 14,400. Its derived subgroup is the largest finite subgroup W(H4)/Z2 of SO(4) of order 7200. Moreover, up to conjugacy, it has five non-normal maximal subgroups of orders 144, two 240, 400 and 576. Two groups [W(H2) × W(H2)]Z4 and W(H3) × Z2 possess non- crystallographic structures with orders 400 and 240 respectively. The groups of orders 144, 240 and 576 are the extensions of the Weyl groups of the root systems of SU(3) × SU(3), SU(5) and SO(8) respectively. We represent the maximal subgroups of W(H4) with sets of quaternion pairs acting on the quaternionic root systems. © 2005 Elsevier Inc. All rights reserved. AMS classification: 20G20; 20F05; 20E34 Keywords: Structure of groups; Quaternions; Coxeter groups; Subgroup structure ∗ Corresponding author. Tel.: +90 342 360 1200; fax: +90 342 360 1013. E-mail address: [email protected] (R. Koc¸). 0024-3795/$ - see front matter ( 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.laa.2005.07.018 442 M. Koca et al. / Linear Algebra and its Applications 412 (2006) 441–452 1. Introduction The non-crystallographic Coxeter group W(H4) of order 14,400 generates some interests [1] for its relevance to the quasicrystallographic structures in condensed matter physics [2,3] as well as its unique relation with the E8 gauge symmetry asso- ciated with the heterotic superstring theory [4]. The Coxeter group W(H4) [5]isthe maximal finite subgroup of O(4), the finite subgroups of which have been classified by du Val [6] and by Conway and Smith [7]. It is also one of the maximal subgroups of the Weyl group W(E8) splitting 240 nonzero roots of E8 into two equal size disjoint sets. One set can be represented by the icosians q (quaternionic elements√ of the binary σq σ = 1− 5 icosahedral group I) and the remaining set is [8,9] where 2 . Embedding of W(H4) in W(E8) is studied in detail in Refs. [10,11]. In this paper we study the maximal subgroups of W(H4) and show that it possesses, up to conjugacy, five maximal subgroups of orders 144, 240, 400 and 576. They cor- respond to the symmetries of certain Dynkin and Coxeter diagrams. Another obvious maximal subgroup W(H4) ≡ W(H4)/Z2 of W(H4) of order 7200 is also the largest finite subgroup of SO(4). Two groups of orders 400 and 240 are related to the symme- H ⊕ H H tries of the non-crystallographic Coxeter graphs 2 2 and 3 respectively. The remaining groups of orders 144, 240 and 576 are associated with the symmetries of A ⊕ A A D the crystallographic root systems 2 2, 4 and 4 respectively (we interchange- ably use A2 ≈ SU(3), A4 ≈ SU(5), D4 ≈ SO(8)). Embeddings of these groups in W(H4) are not trivial and the main objective of this work is to clarify this issue. In the context of quaternionic representation of the H4 root system by the elements of binary icosahedral group, identifications of the quaternionic root systems of the maximal subgroups will be simple. The maximal subgroups of orders 144, 400 and A ⊕ A ,H ⊕ H D 576 associated with the root systems 2 2 2 2 and 4 respectively have close relations with the maximal subgroups of the binary icosahedral group I since it has three maximal subgroups. However the maximal subgroups associated with the Coxeter diagram H3 and the Dynkin diagram A4 do not have such correspondences and some care should be undertaken. In Section 2 we introduce the root system of H4 in terms of the icosions I and identify the maximal subgroups of the binary icosahedral group. We discuss briefly the method as to how the group elements of W(H4) are obtained from icosians. In Section 3, we study in terms of quaternions, the group structure associated with the A ⊕ A graph 2 2. Section 4 is devoted to a similar analysis of the symmetries of the H ⊕ H non-crystallographic root system 2 2. In Section 5 we deal with the extension of W(D4) with a cyclic symmetry of its Dynkin diagram, the group of order 576, which was also studied in a paper of ours [12] in a different context. Section 6 is devoted to the study of the automorphism of the H3 root system and we show that it is, up to conjugacy, a group of order 240 when it acts in the four-dimensional space. In Section 7 we study the somewhat intricate structure of the automorphism group of A4 using quaternions. M. Koca et al. / Linear Algebra and its Applications 412 (2006) 441–452 443 Table 1 Conjugacy classes of the binary icosahedral group I represented by quaternions Conjugacy classes and Elements of the conjugacy classes also denoted by their numbers orders of elements (cyclic permutations in e1,e2,e3 should be added if not included) 11 2 −1 + : 1 (τ ± e ± σe ) 10 12 2 1 3 − : 1 (−τ ± e ± σe ) 5122 1 3 : 1 (σ ± e ± τe ) 10 12+ 2 1 2 : 1 (−σ ± e ± τe ) 512− 2 1 2 + : 1 ( ± e ± e ± e ), 1 ( ± τe ± σe ) 6202 1 1 2 3 2 1 1 2 − : 1 (− ± e ± e ± e ), 1 (− ± τe ± σe ) 3202 1 1 2 3 2 1 1 2 1 15+ : e1,e2,e3, (σ e1 ± τe2 ± e3) 430: 2 − :−e , −e , −e , 1 (−σe ± τe ± e ) 15 1 2 3 2 1 2 3 √ √ τ = 1+ 5 σ = 1− 5 Here 2 and 2 . Fig. 1. The Coxeter diagram of H4 with quaternionic simple roots. 2. H4 with icosians The root system of the Coxeter diagram H4 consists of 120 roots which can be represented by the quaternionic elements of the binary icosahedral group I given in Table 1 (so is called the group of icosians). They can be generated by reflections on the simple roots depicted in Fig. 1. Any real quaternion can be written as q = q0 + q1e1 + q2e2 + q3e3 where qa (a = 0, 1, 2, 3) are real numbers and pure quaternion units1 ei (i = 1, 2, 3) satisfy the well-known relations eiej =−δij + ij k ek (i,j,k = 1, 2, 3), (1) where ij k is the Levi–Civita symbol. The scalar product of two quaternions p and q is defined by (p, q) = 1 (pq¯ + qp),¯ 2 (2) N(q) = (q, q) = qq¯ = q2 + q2 + q2 + q2 which leads to the norm 0 1 2 3 . 1 Any unit quaternion q =−¯q is a pure quaternion, where the quaternion conjugate is q¯ = q0 − q1e1 − q2e2 − q3e3. 444 M. Koca et al. / Linear Algebra and its Applications 412 (2006) 441–452 A ⊕ A a = 1 ( + Fig. 2. Dynkin diagram of 2 2 with the elements of dihedral group of order 12 ( 2 1 τe + σe a¯ = 1 ( − τe − σe ) 1 2), 2 1 1 2 ). The icosians of the binary icosahedral group I are classified in Table 1 accord- ing to the conjugacy classes. Denote by p, q and r any three elements of I. The transformations defined by [p, r]:q → pqr, (3) ∗ [p, r] : q → pqr¯ (4) preserve the norm qq¯ =¯qq as well as leave the set of icosians I intact so that the pairs ∗ [p, r] and [p, r] represent the group elements of W(H4). Since [p, r]=[−p, −r] and [p, −r]=[−p, r] the W(H4) consists of 120 × 120 = 14,400 elements. Details can be found in Ref. [11]. The element [1, 1]∗ acts as a conjugation, [1, 1]∗ : q →¯q, which is the normalizer of the subgroup SO(4) so that O(4) can be written as the semi-direct product O(4) ≈ SO(4)Z2. It follows from this structure that one of the maximal subgroup W(H4) of order 7200 is obviously a maximal finite subgroup of SO(4). The group W(H4) can be represented by the pair [p, r], p, r ∈ I, possessing 42 conjugacy classes. It is easily found from the structure of the conjugacy classes of I in Table 1 that the maximal subgroups of I can be generated by the sets of quaternions2 : 1 ( + τe + σe ), e , dihedral group of order 12 2 1 1 2 3 (5a) : 1 (τ + σe + e ), e , dihedral group of order 20 2 1 2 3 (5b) : 1 ( + e + e + e ), e . binary tetrahedral group of order 24 2 1 1 2 3 3 (5c) These generators are the representative elements of the conjugacy classes of the max- imal subgroups of I. The binary icosahedral group has two quaternionic irreducible representations. If we denote by I the other quaternionic representation it can be obtained from I by interchanging σ ↔ τ in Table 1. We will use both representations in Section 7. 3. The maximal subgroup of order 144 A little exercise shows that the set of elements in Eq.
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