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Acta Crystallographica Section A Foundations and On the subgroup structure of the hyperoctahedral Advances group in six dimensions ISSN 2053-2733 Emilio Zappa,a,c* Eric C. Dykemana,b,c and Reidun Twarocka,b,c Received 17 January 2014 Accepted 7 April 2014 aDepartment of Mathematics, University of York, York, UK, bDepartment of Biology, University of York, York, UK, and cYork Centre for Complex Systems Analysis, University of York, York, UK. Correspondence e-mail: [email protected]
The subgroup structure of the hyperoctahedral group in six dimensions is investigated. In particular, the subgroups isomorphic to the icosahedral group are studied. The orthogonal crystallographic representations of the icosahedral group are classified and their intersections and subgroups analysed, using results from graph theory and their spectra.
1. Introduction denoted by B6, which is a subgroup of Oð6Þ and can be represented in the standard basis of R6 as the set of all 6 6
The discovery of quasicrystals in 1984 by Shechtman et al. has orthogonal and integral matrices. The subgroups of B6 which spurred the mathematical and physical community to develop are isomorphic to the icosahedral group constitute the integral mathematical tools in order to study structures with non- representations of it; among them, the crystallographic ones crystallographic symmetry. are those which split, in GLð6; RÞ, into two three-dimensional Quasicrystals are alloys with five-, eight-, ten- and 12-fold irreducible representations of I. Therefore, they carry two symmetry in their atomic positions (Steurer, 2004), and subspaces in R3 which are invariant under the action of I and therefore they cannot be organized as (periodic) lattices. In can be used to model the quasiperiodic structures. crystallographic terms, their symmetry group G is noncrys- The embedding of the icosahedral group into B6 has been tallographic. However, the noncrystallographic symmetry used extensively in the crystallographic literature. Katz (1989), leaves a lattice invariant in higher dimensions, providing an Senechal (1995), Kramer & Zeidler (1989), Baake & Grimm integral representation of G. If such a representation is (2013), among others, start from a six-dimensional crystal- reducible and contains a two- or three-dimensional invariant lographic representation of I to construct three-dimensional subspace, then it is referred to as a crystallographic repre- Penrose tilings and icosahedral quasicrystals. Kramer (1987) sentation, following terminology given by Levitov & Rhyner and Indelicato et al. (2011) also apply it to study structural (1988). This is the starting point to construct quasicrystals via transitions in quasicrystals. In particular, Kramer considers in the cut-and-project method described by, among others, B6 a representation of I and a representation of the octahe- Senechal (1995), or as a model set (Moody, 2000). dral group O which share a tetrahedral subgroup, and defines In this paper we are interested in icosahedral symmetry. The a continuous rotation (called Schur rotation) between cubic icosahedral group I consists of all the rotations that leave a and icosahedral symmetry which preserves intermediate regular icosahedron invariant, it has size 60 and it is the largest tetrahedral symmetry. Indelicato et al. define a transition of the finite subgroups of SOð3Þ. I contains elements of order between two icosahedral lattices as a continuous path five, therefore it is noncrystallographic in three dimensions; connecting the two lattice bases keeping some symmetry the (minimal) crystallographic representation of it is six- preserved, described by a maximal subgroup of the icosahe- dimensional (Levitov & Rhyner, 1988). The full icosahedral dral group. The rationale behind this approach is that the two group, denoted by I h, also contains the reflections and is equal corresponding lattice groups share a common subgroup. These to I C2, where C2 denotes the cyclic group of order two. I h two approaches are shown to be related (Indelicato et al., is isomorphic to the Coxeter group H3 (Humphreys, 1990) and 2012), hence the idea is that it is possible to study the transi- is made up of 120 elements. In this work, we focus on the tions between icosahedral quasicrystals by considering two icosahedral group I because it plays a central role in appli- distinct crystallographic representations of I in B6 which cations in virology (Indelicato et al., 2011). However, our share a common subgroup. considerations apply equally to the larger group I h. These papers motivate the idea of studying in some detail R6 Levitov & Rhyner (1988) classified the Bravais lattices in the subgroup structure of B6. In particular, we focus on the that are left invariant by I: there are, up to equivalence, subgroups isomorphic to the icosahedral group and its exactly three lattices, usually referred to as icosahedral subgroups. Since the group is quite large (it has 266! elements), Bravais lattices, namely the simple cubic (SC), body-centred we use for computations the software GAP (The GAP Group, cubic (BCC) and face-centred cubic (FCC). The point group 2013), which is designed to compute properties of finite of these lattices is the six-dimensional hyperoctahedral group, groups. More precisely, based on Baake (1984), we generate
Acta Cryst. (2014). A70, 417–428 doi:10.1107/S2053273314007712 417 research papers
the elements of B6 in GAP as a subgroup of the symmetric Table 1 group S and then find the classes of subgroups isomorphic to Character table of the icosahedral group. 12 pffiffiffi the icosahedral group. Among them we isolate, using results Note that ¼ð 5 þ 1Þ=2 is the golden ratio. from character theory, the class of crystallographic repre- sentations of I. In order to study the subgroup structure of this class, we propose a method using graph theory and their spectra. In particular, we treat the class of crystallographic representations of I as a graph: we fix a subgroup G of I and say that two elements in the class are adjacent if their inter- section is equal to a subgroup isomorphic to G. We call the resulting graph G-graph. These graphs are quite large and difficult to visualize; however, by analysing their spectra (Cvetkovic et al., 1995) we can study in some detail their PðBMÞ¼PðBÞ; ðBMÞ¼M 1 ðBÞM; M 2 GLðn; ZÞ: topology, hence describing the intersection and the subgroups shared by different representations. We notice that a change of basis in the lattice leaves the The paper is organized as follows. After recalling, in x2, the point group invariant, whereas the corresponding lattice definitions of point group and lattice group, we define, in x3, groups are conjugated in GLðn; ZÞ. Two lattices are inequi- the crystallographic representations of the icosahedral group valent if the corresponding lattice groups are not conjugated and the icosahedral lattices in six dimensions. We provide, in GLðn; ZÞ (Pitteri & Zanzotto, 2002). following Kramer & Haase (1989), a method for the As a consequence of the crystallographic restriction [see, construction of the projection into three dimensions using for example, Baake & Grimm (2013)] five- and n-fold tools from the representation theory of finite groups. In x4we symmetries, where n is a natural number greater than six, are classify, with the help of GAP, the crystallographic repre- forbidden in dimensions two and three, and therefore any sentations of I.Inx5 we study their subgroup structure, group G containing elements of such orders cannot be the introducing the concept of G-graph, where G is a subgroup point group of a two- or three-dimensional lattice. We there- of I. fore call these groups noncrystallographic. In particular, three- dimensional icosahedral lattices cannot exist. However, a noncrystallographic group leaves some lattices invariant in 2. Lattices and noncrystallographic groups higher dimensions and the smallest such dimension is called Rn R Let bi, i ¼ 1; ...; n be a basis of , and let B 2 GLðn; Þ be the minimal embedding dimension. Following Levitov & the matrix whose columns are the components of bi with Rhyner (1988), we introduce: f ; i ¼ ; ...; ng Rn respect to the canonical basis ei 1 of . A lattice Definition 2.1. Let G be a noncrystallographic group. A in Rn is a Z-free module of rank n with basis B, i.e. crystallographic representation of G is a D-dimensional Pn representation of G such that: Z LðBÞ¼ x ¼ mibi : mi 2 : (1) the characters of are integers; i¼1 (2) is reducible and contains a two- or three-dimensional Any other lattice basis is given by BM, where M 2 GLðn; ZÞ, representation of G. the set of invertible matrices with integral entries (whose We observe that the first condition implies that G must be determinant is equal to 1) (Artin, 1991). the subgroup of the point group of a D-dimensional lattice. The point group of a lattice L is given by all the orthogonal The second condition tells us that contains either a two- transformations that leave the lattice invariant (Pitteri & or three-dimensional invariant subspace E of RD, usually Zanzotto, 2002): referred to as physical space (Levitov & Rhyner, 1988). PðBÞ¼fQ 2 OðnÞ : 9M 2 GLðn; ZÞ s:t: QB ¼ BMg: We notice that, if Q 2PðBÞ, then B 1QB ¼ M 2 GLðn; ZÞ. In other words, the point group consists of all the orthogonal 3. Six-dimensional icosahedral lattices matrices which can be represented in the basis B as integral The icosahedral group I is generated by two elements, g and matrices. The set of all these matrices constitute the lattice 2 g , such that g2 ¼ g3 ¼ðg g Þ5 ¼ e, where e denotes the group of the lattice: 3 2 3 2 3 identity element. It has size 60 and it is isomorphic to A5, the ðBÞ¼fM 2 GLðn; ZÞ : 9Q 2PðBÞ s:t: M ¼ B 1QBg: alternating group of order five (Artin, 1991). Its character table is given in Table 1. The lattice group provides an integral representation of the From the character table we see that the (minimal) crys- point group and these are related via the equation tallographic representation of I is six-dimensional and is given 6 1 I R ðBÞ¼B PðBÞB; by T1 T2. Therefore, leaves a lattice in invariant. Levitov & Rhyner (1988) proved that the three inequivalent and moreover the following hold (Pitteri & Zanzotto, 2002): Bravais lattices of this type, mentioned in the Introduction and
418 Emilio Zappa et al. Subgroup structure of the hyperoctahedral group Acta Cryst. (2014). A70, 417–428 research papers
n referred to as icosahedral (Bravais) lattices, are given by, ImðPiÞ is equal to an ni-dimensional subspace Vi of F invar- respectively: iant under i. In our case, we have two projection operators, R6 R3 Z Pi : ! , i = 1, 2, corresponding to the IRs T1 and T2, È LSC ¼fx ¼ðx1; ...; x6Þ : xi 2 g; É L ¼ x ¼ 1 ðx ; ...; x Þ : x 2 Z; x ¼ x mod2; 8i; j ¼ 1; ...; 6 ; respectively. We assume the image of P1,ImðP1Þ, to be equal to BCC È2 1 6 i i Pj É k ? 6 E , and ImðP Þ¼E .Iffe ; j ¼ 1; ...; 6g is the canonical basis L ¼ x ¼ 1 ðx ; ...; x Þ : x 2 Z; x ¼ 0 mod2 : 2 j FCC 2 1 6 i i¼1 i of R6, then a basis of Ek (respectively E?) can be found
We note that a basis of the SC lattice is the canonical basis considering the set fe^j :¼ Piej; j ¼ 1; ...; 6g for i ¼ 1 6 of R . Its point group is given by (respectively i ¼ 2) and then extracting a basis Bi from it. Since dim Ek =dimE? ¼ 3, we obtain B ¼fe^ ; e^ ; e^ g, for i P ¼fQ 2 Oð6Þ : Q ¼ M 2 GLð6; ZÞg ¼ Oð6Þ\GLð6; ZÞ’Oð6; ZÞ; i i;1 i;2 i;3 SC = 1, 2. The matrix R can be thus written as ð1Þ which is the hyperoctahedral group in dimension six. In the ¼ ð Þ R e|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}^1;1; e^1;2; e^1;3; |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}e^2;1; e^2;2; e^2;3 : 4 following, we will denote this group by B6, following Ek E? Humphreys (1996). We point out that this notation comes basis of basis of k ? from Lie theory: indeed, B6 represents the root system of the Denoting by and the 3 6 matrices which represent Lie algebra soð13Þ (Fulton & Harris, 1991). However, the P1 and P2 in the bases B1 and B2, respectively, we have, by corresponding reflection group WðB6Þ is isomorphic to the linear algebra hyperoctahedral group in six dimensions (Humphreys, 1990). ! k All three lattices have point group B6, whereas their lattice R 1 ¼ : ð5Þ groups are different and, indeed, they are not conjugate in ? GLð6; ZÞ (Levitov & Rhyner, 1988). Since R 1H¼H0R 1 [cf. equation (2)], we obtain Let H be a subgroup of B6 isomorphic to I. H provides a (faithful) integral and orthogonal representation of I. More- k k ? ? ðHðgÞvÞ¼T ð ðvÞÞ; ðHðgÞvÞ¼T ð ðvÞÞ; ð6Þ R 1 2 over, if H’T1 T2 in GLð6; Þ, then H is also crystal- lographic (in the sense of Definition 2.1). All of the other for all g 2Iand v 2 R6. In particular, the following diagram crystallographic representations are given by B 1HB, where commutes B 2 GLð6; RÞ is a basis of an icosahedral lattice in R6. Therefore we can focus our attention, without loss of gener- ality, on the orthogonal crystallographic representations.
3.1. Projection operators Let H be a crystallographic representation of the icosahe- The set ðH; kÞ is the starting point for the construction of dral group. H splits into two three-dimensional irreducible quasicrystals via the cut-and-project method (Senechal, 1995; representations (IRs), T and T ,inGLð6; RÞ. This means that 1 2 Indelicato et al., 2012). there exists a matrix R 2 GLð6; RÞ such that ! T 0 H0 :¼ R 1HR ¼ 1 : ð2Þ 4. Crystallographic representations of I 0 T2 From the previous section it follows that the six-dimensional
The two IRs T1 and T2 leave two three-dimensional hyperoctahedral group B6 contains all the (minimal) ortho- subspaces invariant, which are usually referred to as the gonal crystallographic representations of the icosahedral physical (or parallel) space Ek and the orthogonal space E? group. In this section we classify them, with the help of the (Katz, 1989). In order to find the matrix R (which is not unique computer software programme GAP (The GAP Group, 2013). in general), we follow (Kramer & Haase, 1989) and use results from the representation theory of finite groups (for proofs and 4.1. Representations of the hyperoctahedral group B6 further results see, for example, Fulton & Harris, 1991). In Permutation representations of the n-dimensional hyper- particular, let :G ! GLðn; FÞ be an n-dimensional repre- octahedral group Bn in terms of elements of S2n, the symmetric sentation of a finite group G over a field F (F = R, C). group of order 2n, have been described by Baake (1984). In By Maschke’s theorem, splits, in GLðn; FÞ,as this subsection, we review these results because they allow us m ... m , where : G ! GLðn ; FÞ is an n - 1 1 r r i i i to generate B6 in GAP and further study its subgroup struc- dimensional IR of G. Then the projection operator ture. n ni Pi : F ! F is given by It follows from equation (1) that B consists of all the X 6 ni orthogonal integral matrices. A matrix A ¼ðaijÞ of this kind Pi :¼ ðgÞ ðgÞ; ð3Þ T j j i must satisfy AA ¼ I6, the identity matrix of order six, and G g2I have integral entries only. It is easy to see that these conditions where denotes the complex conjugate of the character of imply that A has entries in {0, 1} and each row and column i the representation i. This operator is such that its image contains 1 or 1 only once. These matrices are called signed
Acta Cryst. (2014). A70, 417–428 Emilio Zappa et al. Subgroup structure of the hyperoctahedral group 419 research papers permutation matrices. It is straightforward to see that any A 2 B6 can be written in the form NQ, where Q is a 6 6 permutation matrix and N is a diagonal matrix with each diagonal entry being either 1 or 1. We can thus associate with each matrix in B6 a pair ða; Þ, Z6 where a 2 2 is a vector given by the diagonal elements of N, and 2 S6 is the permutation associated with Q. The set of all these pairs constitutes a group (called the wreath product Z Z of 2 and S6, and denoted by 2 o S6; Humphreys, 1996) with the multiplication rule given by ða; Þðb; Þ :¼ða þ b; Þ; 2 Figure 1 A planar representation of an icosahedral surface, showing our labelling convention for the where þ2 denotes addition modulo 2 and vertices; the dots represent the locations of the symmetry axes corresponding to the ða Þk :¼ a ðkÞ; a ¼ða1; ...; a6Þ: generators of the icosahedral group and its subgroups. The kite highlighted is a fundamental domain of the icosahedral group. Z o S and B are isomorphic, an isomorphism 2 6 6 ^ T being the following: Using equation (11) we obtain a representation I : I!B6 given by ai ½Tða; Þ ij :¼ð 1Þ ðiÞ;j: ð8Þ 0 1 0 1 000001 000001 6 B C B C It immediately follows that |B6|=26! = 46 080. A set of B 000010C B 000100C B C B C generators is given by B 00 1000C B 0 10000C IIð^ g Þ¼B C; IIð^ g Þ¼B C: 2 B 000 100C 3 B 00 1000C :¼ð0; ð1; 2ÞÞ; :¼ð0; ð1; 2; 3; 4; 5; 6ÞÞ; :¼ðð0; 0; 0; 0; 0; 1Þ; id Þ; @ A @ A S6 010000 100000 ð9Þ 100000 000010 ð12Þ which satisfy the relations We see that ðg Þ¼ 2 and ðg Þ¼0, so that, by looking 2 ¼ 2 ¼ 6 ¼ð0; id Þ: I^ 2 I^ 3 S6 at the character table of I, we have Z o ! Finally, the function ’ : 2 S6 S12 defined by ^ ¼ þ ; ( I T1 T2
ðkÞþ6ak if 1 k 6 which implies, using Maschke’s theorem (Fulton & Harris, ’ða; ÞðkÞ :¼ ^ R ðk 6Þþ6ð1 ak 6Þ if 7 k 12 1991), that II’T1 T2 in GLð6; Þ. Therefore, the subgroup I^ of B is a crystallographic representation of I. ð10Þ 6 Before we continue, we recall the following (Humphreys, Z is injective and maps any element of 2 o S6 into a permutation 1996): of S12, and provides a faithful permutation representation of B6 as a subgroup of S12. Combining equation (8) with the Definition 4.1. Let H be a subgroup of a group G.The inverse of equation (10) we get the function conjugacy class of H in G is the set
1 1 :¼ T : S12 ! B6; ð11Þ CGðHÞ :¼fgHg : g 2 Gg: which can be used to map a permutation into an element of B6. In order to find all the other crystallographic representa- 4.2. Classification tions, we use the following scheme:
In this subsection we classify the orthogonal crystal- (a) we generate B6 as a subgroup of S12 using equations (9) lographic representations of the icosahedral group. We start and (10); by recalling a standard way to construct such a representation, (b) we list all the conjugacy classes of the subgroups of B6 following Zappa et al. (2013). We consider a regular icosahe- and find a representative for each class; dron and we label each vertex by a number from one to 12, so (c) we isolate the classes whose representatives have order that the vertex opposite to vertex i is labelled by i þ 6 (see 60; Fig. 1). This labelling induces a permutation representation (d) we check if these representatives are isomorphic to I;
: I!S12 given by (e) we map these subgroups of S12 into B6 using equation (11) and isolate the crystallographic ones by checking the ðg Þ¼ð1; 6Þð2; 5Þð3; 9Þð4; 10Þð7; 12Þð8; 11Þ; 2 characters; denoting by S the representative, we decompose ðg Þ¼ð1; 5; 6Þð2; 9; 4Þð7; 11; 12Þð3; 10; 8Þ: S 3 as
420 Emilio Zappa et al. Subgroup structure of the hyperoctahedral group Acta Cryst. (2014). A70, 417–428 research papers
¼ m þ m þ m þ m þ m ; þ S 1 A 2 T1 3 T2 4 G 5 H able fact is that they split into two different classes in B6 .To N þ mi 2 ; i ¼ 1; ...; 5: see this, we first need to generate B6 . In particular, with GAP we isolate the subgroups of index two in B , which are normal Note that S is crystallographic if and only if m ¼ m ¼ 1 6 2 3 in B , and then, using equation (11), we find the one whose and m ¼ m ¼ m ¼ 0. 6 1 4 5 generators have determinant equal to one. In particular, we We implemented steps (1)–(4) in GAP (see Appendix C). have There are three conjugacy classes of subgroups isomorphic to þ I in B6. Denoting by Si ¼hg2;i; g3;ii the representatives of the B6 ¼hð1; 2; 6; 4; 3Þð7; 8; 12; 10; 9Þ; ð5; 11Þð6; 12Þ; classes returned by GAP, we have, using equation (11), ð1; 2; 6; 5; 3Þð7; 8; 12; 11; 9Þ; ð5; 12; 11; 6Þi: ðg Þ¼2; ðg Þ¼3 ) ¼ 2 þ ) S ’ 2A G; S1 2;1 S1 3;1 S1 A G 1 We can then apply the same procedure to find the crystal- ðg Þ¼ 2; ðg Þ¼0 ) ¼ þ ) S ’ T T ; I S2 2;2 S2 3;2 S2 T1 T2 2 1 2 lographic representations of , and see that they split into two classes, each one of size 96. Again we can choose I^ as a S ðg2;3Þ¼2; S ðg3;3Þ¼0 ) S ¼ A þ H ) S3 ’ A H: 3 3 3 representative for one of these classes; a representative K^ for Since 2A is decomposable into two one-dimensional the other one is given by representations, it is not strictly speaking two dimensional in 0 1 0 1 010000 000100 the sense of Definition 2.1, and as a consequence, only the B C B C *B 100000C B 100000C+ second class contains the crystallographic representations of B C B C B 00 1000C B 00000 1 C I. A computation in GAP shows that its size is 192. We thus KK¼^ B C; B C : have the following: B 000001C B 010000C @ 0000 10A @ 00 1000A Proposition 4.1. The crystallographic representations of I in 000100 000010 B 6 form a unique conjugacy class in the set of all the classes of ð13Þ subgroups of B6, and its size is equal to 192. We note that in the more general case of I h,wecan We briefly point out that the other two classes of subgroups construct the crystallographic representations of I starting I h isomorphic to in B6 have an interesting algebraic intepre- from the crystallographic representations of I. First of all, we tation. First of all, we observe that B6 is an extension of S6, recall that I ¼ I C , where C is the cyclic group of order since according to Humphreys (1996): h 2 2 two. Let H be a crystallographic representation of I in B6, and Z6 Z Z6 ¼f ; g B6= 2 ’ð 2 o S6Þ= 2 ’ S6: let 1 1 be a one-dimensional representation of C2. Then the representation H^ given by Following Janusz & Rotman (1982), it is possible to embed ^ the symmetric group S5 into S6 in two different ways. The H :¼H ; canonical embedding is achieved by fixing a point in f1; ...; 6g where denotes the tensor product of matrices, is a repre- and permuting the other five, whereas the other embedding is sentation of I in B and it is crystallographic in the sense of by means of the so-called ‘exotic map’ ’ : S ! S , which acts h 6 5 6 Definition 2.1 (Fulton & Harris, 1991). on the six 5-Sylow subgroups of S5 by conjugation. Recalling that the icosahedral group is isomorphic to the alternating 4.3. Projection into the three-dimensional space group A , which is a normal subgroup of S , then the canonical 5 5 We study in detail the projection into the physical space Ek embedding corresponds to the representation 2A G in B6, while the exotic one corresponds to the representation A H. using the methods described in x3.1. ^ In what follows, we will consider the subgroup I^ previously Let I be the crystallographic representation of I given in defined as a representative of the class of the crystallographic equation (12). Using equation (3) with ni ¼ 3 and jGj¼ representations of I, and denote this class by C ðIIÞ^ . jIj ¼ 60 we obtain the following projection operators B6 0 pffiffiffi 1 Recalling that two representations Dð1Þ and Dð2Þ of a group 5 p1ffiffiffi 1 111 G are said to be equivalent if there are related via a similarity B C B 1 5 p1ffiffiffi 1 11C transformation, i.e. there exists an invertible matrix S such that B C 1 ffiffiffiB 11 5 p1ffiffiffi 11C P1 ¼ p B C; ð1Þ ð2Þ 1 2 5B 1 11 5 11ffiffiffi C D ¼ SD S ; @ p A 1 1 11 5 p1ffiffiffi then an immediate consequence of Proposition 4.1 is the 111115 following:
Corollary 4.1. Let H1 and H2 be two orthogonal crystal- 0 pffiffiffi 1 lographic representations of I.ThenH and H are equiva- 5 111ffiffiffi 1 1 1 2 B p C lent in B . B 1 5 111ffiffiffi 1 C 6 B p C 1 ffiffiffiB 1 1 5 p 11ffiffiffi 1 C We observe that the determinant of the generators of I^ in P2 ¼ p B C: 2 5B 11 1 5 1ffiffiffi 1 C equation (12) is equal to one, so that II2^ Bþ :¼ @ p A 6 111 1 5 p 1ffiffiffi fA 2 B6 : detA ¼ 1g. Proposition 4.1 implies that all the þ 1 1 1 1 1 5 crystallographic representations belong to B6 . The remark-
Acta Cryst. (2014). A70, 417–428 Emilio Zappa et al. Subgroup structure of the hyperoctahedral group 421 research papers
Table 2 Table 4 ^ Explicit forms of the IRs T1 and T2 with II’T1 T2. Permutation representations of the generators of the subgroups of the icosahedral group. Generator Irrep T1 Irrep T2 ðg2Þ¼ð1; 6Þð2; 5Þð3; 9Þð4; 10Þð7; 12Þð8; 11Þ 0 1 0 1 ðg2dÞ¼ð1; 12Þð2; 8Þð3; 4Þð5; 11Þð6; 7Þð9; 10Þ 11 1 1 ðg3Þ¼ð1; 5; 6Þð2; 9; 4Þð7; 11; 12Þð3; 10; 8Þ g 1 @ A 1 @ A 2 1 1 1 1 ðg3dÞ¼ð1; 10; 2Þð3; 5; 12Þð4; 8; 7Þð6; 9; 11Þ 2 2 ð Þ¼ð Þð Þ 1 1 1 1 g5 1; 2; 3; 4; 5 7; 8; 9; 10; 11 ðg5dÞ¼ð1; 10; 11; 3; 6Þð4; 5; 9; 12; 7Þ
0 1 0 1 11 11 Table 5 g3 1 @ A 1 @ A 1 1 1 1 Sizes of the classes of subgroups of the icosahedral group in I and B6. 2 2 1 1 11 Subgroup jC ðGÞj jC ðK Þj I B6 G T 5 480 D10 6 576 D6 10 960 Table 3 D4 5 120 Nontrivial subgroups of the icosahedral group. C5 6 576 C3 10 320 T D stands for the tetrahedral group, 2n for the dihedral group of size 2n, and C2 15 180 Cn for the cyclic group of size n. Subgroup Generators Relations Size
2 3 3 dimensions centred at the origin (Senechal, 1995; Katz, 1989; T g2; g3d g2 ¼ g3d ¼ðg2g3dÞ ¼ e 12 2 5 2 Indelicato et al., 2011). D10 g2d; g5d g2d ¼ g5d ¼ðg5dg2dÞ ¼ e 10 2 3 2 Let K be another crystallographic representation of I in B . D6 g2d; g3 g2d ¼ g3 ¼ðg3g2dÞ ¼ e 6 6 5 By Proposition 4.1, K and I^ are conjugated in B . Consider C5 g5d g5d ¼ e 5 6 2 2 2 ^ 1 D4 g2d; g2 g2d ¼ g2 ¼ðg2g2dÞ ¼ e 4 M 2 B6 such that MIM ¼Kand let S ¼ MR. We have C g g3 ¼ e 3 3 3 3 1 1 1 1 1 ^ 2 S KS ¼ðMRÞ KðMRÞ¼R M KMR ¼ R IR ¼ T1 T2: C2 g2 g2 ¼ e 2 Therefore it is possible, with a suitable choice of the reducing matrices, to project all the crystallographic representations of
The rank of these operators is equal to three. We choose as a I in B6 in the same physical space. basis of Ek and E? the following linear combination of the columns c of the projection operators P , for i = 1, 2 and i;j i 5. Subgroup structure j ¼ 1; ...; 6: ! The nontrivial subgroups of I are listed in Table 3, together c þ c c c c þ c c c c þ c c c with their generators (Hoyle, 2004). Note that T , D and D 1;1 1;5 ; 1;2 1;4 ; 1;3 1;6; 2;1 2;5 ; 2;2 2;4 ; 2;3 2;6 : 10 6 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}2 2 2 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}2 2 2 are maximal subgroups of I, and that D4, C5 and C3 are k ? basis of E basis of E normal subgroups of T , D10 and D6, respectively (Humphreys, 1996; Artin, 1991). The permutation representations of the With a suitable rescaling, we obtain the matrix R given by generators in S are given in Table 4 (see also Fig. 1). 0 1 12 10 01 Since I is a small group, its subgroup structure can be easily B C obtained in GAP by computing explicitly all its conjugacy B 0 1 1 0 C B C classes of subgroups. In particular, there are seven classes of 1 B 10 0 1 C R ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B C: nontrivial subgroups in I: any subgroup H of I has the 2ð2 þ Þ B 0 11 0 C @ 10 01A property that, if K is another subgroup of I isomorphic to H, 10 0 1 then H and K are conjugate in I (this property is referred to as the ‘friendliness’ of the subgroup H; Soicher, 2006). In other The matrix R is orthogonal and reduces I^ as in equation (2). words, denoting by nG the number of subgroups of I In Table 2 we give the explicit forms of the reduced repre- isomorphic to G, i.e. sentation. The matrix representation in Ek of P is given by 1 nG :¼jfH< I : H ’ Ggj; ð14Þ [see equation (5)] 0 1 we have (cf. Definition 4.1) 0 10 1 1 n ¼jC ðGÞj: k ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi @ 1 0 10A: G I 2ð2 þ Þ 01 10 In Table 5 we list the size of each class of subgroups in I. Geometrically, different copies of C2, C3 and C5 correspond to k The orbit fT1ð ðejÞÞg, where fej; j ¼ 1; ...; 6g is the cano- the different two-, three- and fivefold axes of the icosahedron, R6 nical basis of , represents a regular icosahedron in three respectively. In particular, different copies of D10 stabilize one
422 Emilio Zappa et al. Subgroup structure of the hyperoctahedral group Acta Cryst. (2014). A70, 417–428 research papers
of the six fivefold axes of the icosahedron, and each copy of D6 crystallographic representations of I may share subgroups. In stabilizes one of the ten threefold axes. Moreover, it is possible order to describe more precisely the subgroup structure of to inscribe five tetrahedra into a dodecahedron, and each C ðIIÞ^ we will use some basic results from graph theory and B6 different copy of the tetrahedral group in I stabilizes one of their spectra, which we are going to recall in the next section. these tetrahedra. 5.2. Some basic results of graph theory and their spectra 5.1. Subgroups of the crystallographic representations of I In this section we recall, without proofs, some concepts and Let G be a subgroup of I. The function (11) provides a results from graph theory and spectral graph theory. Proofs representation of G in B6, denoted by KG, which is a subgroup and further results can be found, for example, in Foulds (1992) of I^. Let us denote by C ðK Þ the conjugacy class of K in B . and Cvetkovic et al. (1995). B6 G G 6 The next lemma shows that this class contains all the Let G be a graph with vertex set V ¼fv1; ...; vng.The subgroups of the crystallographic representations of I in B6. number of edges incident with a vertex v is called the degree of v. If all vertices have the same degree d, then the graph is Lemma 5.1. Let H 2C ðIIÞ^ be a crystallographic repre- i B6 called regular of degree d. A walk of length l is a sequence of l sentation of I in B6 and let Ki Hi be a subgroup of Hi consecutive edges and it is called a path if they are all distinct. isomorphic to G. Then K 2C ðK Þ. i B6 G A circuit is a path starting and ending at the same vertex and the girth of the graph is the length of the shortest circuit. Two Proof. Since H 2 C ðIIÞ^ , there exists g 2 B such that vertices p and q are connected if there exists a path containing i B6 6 1 ^ 1 0 ^ p and q. The connected component of a vertex v is the set of all gHig ¼ I, and therefore gKig ¼K is a subgroup of I isomorphic to G. Since all these subgroups are conjugate in vertices connected to v. I^ [they are ‘friendly’ in the sense intended by Soicher The adjacency matrix A of G is the n n matrix A ¼ðaijÞ ^ 0 1 whose entries a are equal to one if the vertex v is adjacent to (2006)], there exists h 2 I such that hK h ¼KG. Thus ij i 1 ðhgÞK ðhgÞ ¼K , implying that K 2C ðK Þ. & the vertex vj, and zero otherwise. It can be seen immediately i G i B6 G from its definition that A is symmetric and aii ¼ 0 for all i,so that TrðAÞ¼0. It follows that A is diagonalisable and all its We next show that every element of C ðK Þ is a subgroup B6 G eigenvalues are real. The spectrum of the graph is the set of all of a crystallographic representation of I. the eigenvalues of its adjacency matrix A, usually denoted by ðAÞ. Lemma 5.2.LetK 2C ðK Þ. There exists H 2C ðIIÞ^ i B6 G i B6 such that Ki is a subgroup of Hi. Theorem 5.1. Let A be the adjacency matrix of a graph G
with vertex set V ¼fv1; ...; vng. Let Nkði; jÞ denote the Proof.SinceK 2C ðK Þ, there exists g 2 B such that number of walks of length k starting at vertex vi and finishing i B6 G 6 1 1 ^ at vertex v . We have gKig ¼KG. We define Hi :¼ g Ig. It can be seen imme- j diately that Ki is a subgroup of Hi. & k Nkði; jÞ¼Aij:
As a consequence of these lemmata, C ðK Þ contains all We recall that the spectral radius of a matrix A is defined by B6 G the subgroups of B6 which are isomorphic to G and are ðAÞ :¼ maxfj j : 2 ðAÞg.IfA is a non-negative matrix, i.e. subgroups of a crystallographic representation of I.The if all its entries are non-negative, then ðAÞ2 ðAÞ (Horn & explicit forms of KG are given in Appendix B. We point out Johnson, 1985). Since the adjacency matrix of a graph is non- that it is possible to find subgroups of B6 isomorphic to a negative, j j ðAÞ :¼ r, where 2 ðAÞ and r is the largest subgroup G of I which are not subgroups of any crystal- eigenvalue. r is called the index of the graph G. lographic representation of I. For example, the following subgroup Theorem 5.2. Let 1; ...; n be the spectrum of a graph G, 0 1 0 1 and let r denote its index. Then G is regular of degree r if and 100000 00 1000 only if B C B C *B 0 10000C B 100000C+ B C B C 1 Xn ^ B 00 1000C B 0 10000C 2 TT¼ B C; B C i ¼ r: B 000 100C B 000001C n i¼1 @ 0000 10A @ 000100A 000001 000010 Moreover, if G is regular the multiplicity of its index is equal to the number of its connected components. is isomorphic to the tetrahedral group T ; a computation in GAP shows that it is not a subgroup of any elements in C ðIIÞ^ . B6 Indeed, the two classes of subgroups, C ðK Þ and C ðTTÞ^ , are 5.3. Applications to the subgroup structure B6 T B6 disjoint. Let G be a subgroup of I. In the following we represent the Using GAP, we compute the size of each C ðK Þ (see Table subgroup structure of the class of crystallographic repre- B6 G 5). We observe that jC ðK Þj Acta Cryst. (2014). A70, 417–428 Emilio Zappa et al. Subgroup structure of the hyperoctahedral group 423 research papers H ; H 2C ðIIÞ^ are adjacent to each other (i.e. connected by We notice that, using Theorem 5.2, not all the graphs are 1 2 B6 an edge) in the graph if there exists P 2C ðK Þ such that connected. In particular, the D - and the D -graphs are made B6 G 10 6 P ¼H1 \H2. We can therefore consider the graph up of six connected components, whereas the C3- and the C2- G ¼ðC ðIIÞ^ ; EÞ, where an edge e 2 E is of the form ðH ; H Þ. graphs consist of two connected components. With GAP,we B6 1 2 We call this graph G-graph. implemented a breadth-first search algorithm (Foulds, 1992), Using GAP, we compute the adjacency matrices of the which starts from a vertex i and then ‘scans’ for all the vertices G-graphs. The algorithms used are shown in Appendix C. The connected to it, which allows us to find the connected spectra of the G-graphs are given in Table 6. We first of all components of a given G-graph (see Appendix C). We find notice that the adjacency matrix of the C5-graph is the null that each connected component of the D10-andD6-graphs is matrix, implying that there are no two representations sharing made up of 32 vertices, while for the C3- and C2-graphs each precisely a subgroup isomorphic to C5, i.e. not a subgroup component consists of 96 vertices. For all other subgroups, the containing C5. We point out that, since the adjacency matrix of corresponding G-graph is connected and the connected the D10-graph is not the null one, then there exist crystal- component contains trivially 192 vertices. lographic representations, say Hi and Hj, sharing a maximal We now consider in more detail the case when G is a subgroup isomorphic to D . Since C is a (normal) subgroup maximal subgroup of I. Let H2C ðIIÞ^ and let us consider its 10 5 B6 of D10, then Hi and Hj do share a C5 subgroup, but also a C2 vertex star in the corresponding G-graph, i.e. subgroup. In other words, if two representations share a fivefold axis, then necessarily they also share a twofold axis. VðHÞ :¼fK2C ðIIÞ^ : K is adjacent to Hg: ð15Þ B6 A straightforward calculation based on Theorem 5.2 leads to the following A comparison of Tables 5 and 6 shows that dG ¼ nG [i.e. the Proposition 5.1. Let G be a subgroup of I. Then the number of subgroups isomorphic to G in I, cf. equation (14)] corresponding G-graph is regular. and therefore, since the graph is regular, jVðHÞj ¼ dG ¼ nG. This suggests that there is a one-to-one correspondence between elements of the vertex star of H and subgroups of H In particular, the degree dG of each G-graph is equal to the largest eigenvalue of the corresponding spectrum. As a isomorphic to G; in other words, if we fix any subgroup P of H consequence we have the following: isomorphic to G, then P ‘connects’ H with exactly another representation K. We thus have the following: Proposition 5.2. Let H be a crystallographic representation of I in B6. Then there are exactly dG representations Proposition 5.3. Let G be a maximal subgroup of I. Then for K 2C ðIIÞ^ such that j B6 every P 2C ðK Þ there exist exactly two crystallographic B6 G representations of I, H ; H 2C ðIIÞ^ , such that P ¼H \H . H\Kj ¼ Pj; 9 Pj 2CðKGÞ; j ¼ 1; ...; dG: 1 2 B6 1 2 In particular, we have dG = 5, 6, 10, 0, 30, 20, 60 and 60 for In order to prove it, we first need the following lemma: G¼T; D10; D6, C5; D4; C3; C2 and feg, respectively. Lemma 5.3. Let G be a maximal subgroup of I. Then the In particular, this means that for any crystallographic corresponding G-graph is triangle-free, i.e. it has no circuits of representation of I there are precisely dG other such repre- length three. sentations which share a subgroup isomorphic to G. In other words, we can associate to the class C ðIIÞ^ the ‘subgroup Proof. Let A be the adjacency matrix of the G-graph. By B6 G 3 matrix’ S whose entries are defined by Theorem 5.1, its third power AG determines the number of walks of length three, and in particular its diagonal entries, Sij ¼jHi \Hjj; i; j ¼ 1; ...; 192: 3 ðAGÞii, for i ¼ 1; ...; 192, correspond to the number of trian- The matrix S is symmetric and Sii ¼ 60, for all i, since the gular circuits starting and ending in vertex i.Adirect 3 order of I is 60. It follows from Proposition 5.2 that each row computation shows that ðAGÞii ¼ 0, for all i, thus implying the & of S contains dG entries equal to jGj. Moreover, a rearrange- non-existence of triangular circuits in the graph. ment of the columns of S shows that the 192 crystallographic representations of I can be grouped into 12 sets of 16 such that any two of these representations in such a set of 16 share a Proof of Proposition 5.3.IfP 2C ðK Þ, then, using Lemma B6 G D -subgroup. This implies that the corresponding subgraph of 5.2, there exists H 2C ðIIÞ^ such that P is a subgroup of H . 4 1 B6 1 the D4-graph is a complete graph, i.e. every two distinct Let us consider the vertex star VðH1Þ. We have jVðH1Þj ¼ dG; vertices are connected by an edge. From a geometric point of we call its elements H ; ...; H . Let us suppose that P is 2 dGþ1 view, these 16 representations correspond to ‘six-dimensional not a subgroup of any Hj, for j ¼ 2; ...; dG þ 1. This implies icosahedra’. This ensemble of 16 such icosahedra embedded that P does not connect H1 with any of these Hj. However, into a six-dimensional hypercube can be viewed as a six- since H1 has exactly nG different subgroups isomorphic to G, dimensional analogue of the three-dimensional ensemble of then at least two vertices in the vertex star, say H2 and H3, are five tetrahedra inscribed into a dodecahedron, sharing pair- connected by the same subgroup isomorphic to G, which we wise a C3-subgroup. denote by Q. Therefore we have 424 Emilio Zappa et al. Subgroup structure of the hyperoctahedral group Acta Cryst. (2014). A70, 417–428 research papers Q ¼H1 \H2; Q ¼H1 \H3 ) Q ¼H2 \H3: Table 6 Spectra of the G-graphs for G a nontrivial subgroup of I and G¼feg,the trivial subgroup consisting of only the identity element e. This implies that H1, H2 and H3 form a triangular circuit in the graph, which is a contradiction due to Lemma 5.3, hence the The numbers highlighted are the indices of the graphs, and correspond to their result is proved. & degrees dG. T -graph D10-graph D6-graph C5-graph þ Eig. Mult. Eig. Mult. Eig. Mult. Eig. Mult. It is noteworthy that the situation in B6 is different. If we denote by X1 and X2 the two disjoint classes of crystal- 5 1 6 6 10 6 0 192 þ 345290290 lographic representations of I in B6 [cf. equation (13)], we G 345 290 290 can build, in the same way as described before, the -graphs 150 66 10 6 for X1 and X2, for G¼T; D10 and D6. The result is that the 150 adjacency matrices of all these six graphs are the null matrix of 51 dimension 96. This implies that these graphs have no edges, and so the representations in each class do not share any D4-graph C3-graph C2-graph feg-graph maximal subgroup of I. As a consequence, we have the Eig. Mult. Eig. Mult. Eig. Mult. Eig. Mult. following: 30 1 20 2 60 2 60 1 Proposition 5.4. Let H; K2C ðIIÞ^ be two crystallographic 18 5 4 90 4 90 12 5 B6 12 5 4 100 490 490 representations of I, and P ¼H\K, P 2C ðK Þ, where G is B6 G 615 12 10 490 a maximal subgroup of I. Then H and K are not conjugated in 245 12 5 Bþ. In other words, the elements of B which conjugate H with 031 60 1 6 6 230 K are matrices with determinant equal to 1. 445 815 We conclude by showing a computational method which combines the result of Propositions 4.1 and 5.2. We first recall 0 1 0 1 the following: 0000 10 0000 10 B C B C Definition 5.1. Let H be a subgroup of a group G.The *B 000100C B 00 1000C+ B C B C normaliser of H in G is given by B 00 1000C B 000001C H0 ¼ B C; B C : B 010000C B 100000C 1 @ A @ A NGðHÞ :¼fg 2 G : gHg ¼ Hg: 100000 000 100 00000 1 0 10000 A direct computation shows that the matrix Corollary 5.1. Let H and K be two crystallographic repre- 0 1 100000 sentations of I in B6 and P 2CðKGÞ such that P ¼H\K. Let B C B 0 10000C B C ¼f 2 H 1 ¼Kg B 001000C AH;K M B6 : M M M ¼ B C B 000100C @ 000010A be the set of all the elements of B6 which conjugate H with K and let N ðPÞ be the normaliser of P in B .Wehave 000001 B6 6 belongs to N ðK Þ and conjugate I^ with H . Note that AH;K \ NB ðHÞ 6¼;: B6 D10 0 6 det M ¼ 1. In other words, it is possible to find a nontrivial element M 2 B in the normaliser of P in B which conjugates H with 6 6 6. Conclusions K. In this work we explored the subgroup structure of the hyperoctahedral group in six dimensions. In particular we Proof. Let us suppose A \ N ðHÞ¼;. Then H;K B6 found the class of the crystallographic representations of the MPM 1 6¼ P, for all M 2 A . This implies, since H;K icosahedral group, whose size is 192. Any such representation, MHM 1 ¼K, that P is not a subgroup of K, which is a together with its corresponding projection operator k, can be contradiction. & chosen to construct icosahedral quasicrystals via the cut-and- project method. We then studied in detail the subgroup We give now an explicit example. We consider the repre- structure of this class. For this, we proposed a method based on sentation I^ as in equation (12), and its subgroup K (the spectral graph theory and introduced the concept of G-graph, D10 explicit form is given in Appendix B). With GAP, we find the for a subgroup G of the icosahedral group. This allowed us to other representation H 2CðIIÞ^ such that K ¼ II\H^ . Its study the intersection and the subgroups shared by different 0 D10 0 explicit form is given by representations. We have shown that, if we fix any repre- Acta Cryst. (2014). A70, 417–428 Emilio Zappa et al. Subgroup structure of the hyperoctahedral group 425 research papers sentation H in the class and a maximal subgroup P of H, then Cyclic group C5 [ ¼ expð2 i=5Þ]: there exists exactly one other representation K in the class such that P ¼H\K. As explained in the Introduction, this can be used to describe transitions which keep intermediate symmetry encoded by P. In particular, this result implies in this context that a transition from a structure arising from H via projection will result in a structure obtainable for K via projection if the transition has intermediate symmetry described by P. Therefore, this setting is the starting point to Dihedral group D4 (the Klein Four Group): analyse structural transitions between icosahedral quasicrys- tals, following the methods proposed in Kramer (1987), Katz (1989) and Indelicato et al. (2012), which we are planning to address in a forthcoming publication. These mathematical tools also have many applications in other areas. A prominent example is virology. Viruses package their genomic material into protein containers with regular Cyclic group C3 [! ¼ expð2 i=3Þ]: structures that can be modelled via lattices and group theory. Structural transitions of these containers, which involve rear- rangements of the protein lattices, are important in rendering certain classes of viruses infective. As shown in Indelicato et al. (2011), such structural transitions can be modelled using projections of six-dimensional icosahedral lattices and their Cyclic group C2: symmetry properties. The results derived here therefore have a direct application to this scenario, and the information on the subgroup structure of the class of crystallographic repre- sentations of the icosahedral group and their intersections provides information on the symmetries of the capsid during the transition. APPENDIX B Here we show the explicit forms of KG, the representations in B6 of the subgroups of I, together with their decomposi- APPENDIX A tions in GLð6; RÞ. In order to render this paper self-contained, we provide the 0 1 0 1 character tables of the subgroups of the icosahedral group, 000 001 010000 B C B C following Artin (1991), Fulton & Harris (1991) and Jones *B 000 010C B 000 100C+ B C B C (1990). B 00 1000C B 00000 1 C KT ¼ B C; B C ; Tetrahedral group T [! ¼ expð2 i=3Þ]: B 000 100C B 1000 0 0C @ 010 000A @ 001000A 100 000 0000 10 0 1 0 1 00000 1 000001 B C B C *B 0 1000 0C B 010000C+ Dihedral group D : B C B C 10 B 00010 0C B 0000 10C KD ¼ B C; B C ; 10 B 00100 0C B 100000C @ 0000 10A @ 000100A 10000 0 001000 0 1 0 1 Dihedral group D (isomorphic to the symmetric group S ): 00000 1 000001 6 3 B C B C *B 0 1000 0C B 000100C+ B C B C B 000100C B 0 10000C KD ¼ B C; B C ; 6 B 001000C B 00 1000C @ 0000 10A @ 100000A 100000 000010 426 Emilio Zappa et al. Subgroup structure of the hyperoctahedral group Acta Cryst. (2014). A70, 417–428 research papers 0 1 000001 B C *B 010000C+ B C B 0000 10C KC ¼ B C ; 5 B 100000C @ 000100A 001000 0 1 0 1 00000 1 000 001 B C B C *B 0 1000 0C B 000 010C+ B C B C B 000100C B 00 1000C KD ¼ B C; B C : 4 B 001000C B 000 100C @ 0000 10A @ 010 000A 100000 100 000 0 1 0 0 0001 B C *B 0 0 0100C+ B C B 0 10000C KC ¼ B C ; 3 B 00 1000C @ 1 0 0000A 0 0 0010 0 1 000 001 B C *B 000 010C+ B C B 00 1000C KC ¼ B C : 2 B 000 100C @ A Figure 2 010 000 Algorithm 1. 100 000 K ’ 2T; K ’ 2A E E ; K ’ 2A 2E; K ’ 2A E E ; T D10 2 1 2 D6 2 C5 1 2 K ’ 2B 2B 2B ; K ’ 2A 2E; K ’ 2A 4B: D4 1 2 3 C3 C2 APPENDIX C In this Appendix we show our algorithms, which have been implemented in GAP and used in various sections of the paper. We list them with a number from 1 to 5. Algorithm 1 (Fig. 2): Classification of the crystallographic representations of I (see x4). The algorithm carries out steps Figure 3 1–4 used to prove Proposition 4.1. In the GAP computation, Algorithm 2. the class C ðIIÞ^ is indicated as CB6s60. Its size is 192. B6 Algorithm 2 (Fig. 3): Computation of the vertex star of a given vertex i in the G-graphs. In the following, H stands for the class C ðIIÞ^ of the crystallographic representations of I, B6 i 2f1; ...; 192g denotes a vertex in the G-graph corresponding to the representation H½i and n stands for the size of G: we can use the size instead of the explicit form of the subgroup since, in the case of the icosahedral group, all the non isomorphic subgroups have different sizes. Algorithm 3 (Fig. 4): Computation of the adjacency matrix of the G-graph. Algorithm 4 (Fig. 5): This algorithm carries out a breadth- first search strategy for the computation of the connected component of a given vertex i of the G-graph. Algorithm 5 (Fig. 6): Computation of all connected Figure 4 components of a G-graph. Algorithm 3. Acta Cryst. (2014). A70, 417–428 Emilio Zappa et al. Subgroup structure of the hyperoctahedral group 427 research papers Figure 6 Algorithm 5. Hoyle, R. (2004). Physica D, 191, 261–281. Humphreys, J. (1990). Reflection Groups and Coxeter Groups. Cambridge University Press. Humphreys, J. (1996). A Course in Group Theory. Oxford University Press. Indelicato, G., Cermelli, P., Salthouse, D., Racca, S., Zanzotto, G. & Twarock, R. (2011). J. Math. Biol. 64, 745–773. Figure 5 Indelicato, G., Keef, T., Cermelli, P., Salthouse, D., Twarock, R. & Algorithm 4. Zanzotto, G. (2012). Proc. R. Soc. A, 468, 1452–1471. Janusz, G. & Rotman, J. (1982). Am. Math. Mon. 89, 407–410. We would like to thank Silvia Steila, Pierre-Philippe Jones, H. (1990). Groups, Representations and Physics. Institute of Dechant, Paolo Cermelli and Giuliana Indelicato for useful Physics Publishing. discussions, and Paolo Barbero and Alessandro Marchino for Katz, A. (1989). 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Cvetkovic, D., Doob, M. & Sachs, H. (1995). Spectra of Graphs. Senechal, M. (1995). Quasicrystals and Geometry. Cambridge Heidelberg, Leipzig: Johann Ambrosius Barth. University Press. Foulds, L. (1992). Graph Theory Applications. New York: Springer- Soicher, L. (2006). Oberwolfach Rep. 3, 1809–1811. Report 30/2006. Verlag. Steurer, W. (2004). Z. Kristallogr. 219, 391–446. Fulton, W. & Harris, J. (1991). Representation Theory: A First Course. The GAP Group (2013). GAP – Groups, Algorithms, and Program- Springer-Verlag. ming, Version 4.7.2. http://www.gap-system.org. Horn, R. & Johnson, C. (1985). Matrix Analysis. Cambridge Zappa, E., Indelicato, G., Albano, A. & Cermelli, P. (2013). Int. J. University Press. Non-Linear Mech. 56, 71–78. 428 Emilio Zappa et al. Subgroup structure of the hyperoctahedral group Acta Cryst. (2014). A70, 417–428