On the subgroup structure of the hyperoctahedral group in six dimensions

White Rose Research Online URL for this paper:

http://eprints.whiterose.ac.uk/83826/

Version: Published Version

Article:

Zappa, Emilio, Dykeman, Eric C and Twarock, Reidun orcid.org/0000-0002-1824-2003

(2014) On the subgroup structure of the hyperoctahedral group in six dimensions. Acta

crystallographica. Section A, Foundations and advances. pp. 417-428. ISSN 2053-2733

https://doi.org/10.1107/S2053273314007712

[email protected] https://eprints.whiterose.ac.uk/

Reuse

Items deposited in White Rose Research Online are protected by copyright, with all rights reserved unless indicated otherwise. They may be downloaded and/or printed for private study, or other acts as permitted by national copyright laws. The publisher or other rights holders may allow further reproduction and re-use of the full text version. This is indicated by the licence information on the White Rose Research Online record for the item.

Takedown

If you consider content in White Rose Research Online to be in breach of UK law, please notify us by

Acta Crystallographica Section A Foundations and Advances ISSN 2053-2733

Received 17 January 2014 Accepted 7 April 2014

On the subgroup structure of the hyperoctahedral

group in six dimensions

Emilio Zappa,a,c* Eric C. Dykemana,b,cand Reidun Twarocka,b,c

a

Department of Mathematics, University of York, York, UK,bDepartment of Biology, University of York, York, UK, andc_{York 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

The discovery of quasicrystals in 1984 by Shechtmanet al.has spurred the mathematical and physical community to develop mathematical tools in order to study structures with non-crystallographic symmetry.

Quasicrystals are alloys with five-, eight-, ten- and 12-fold symmetry in their atomic positions (Steurer, 2004), and therefore they cannot be organized as (periodic) lattices. In crystallographic terms, their symmetry group G is noncrys-tallographic. However, the noncrystallographic symmetry leaves a lattice invariant in higher dimensions, providing an integral representation of G. If such a representation is reducible and contains a two- or three-dimensional invariant subspace, then it is referred to as a crystallographic repre-sentation, following terminology given by Levitov & Rhyner (1988). This is the starting point to construct quasicrystalsvia the cut-and-project method described by, among others, Senechal (1995), or as a model set (Moody, 2000).

In this paper we are interested in icosahedral symmetry. The icosahedral group _I consists of all the rotations that leave a regular icosahedron invariant, it has size 60 and it is the largest of the finite subgroups ofSO_ð3_Þ._Icontains elements of order five, therefore it is noncrystallographic in three dimensions; the (minimal) crystallographic representation of it is six-dimensional (Levitov & Rhyner, 1988). The full icosahedral group, denoted by_Ih, also contains the reflections and is equal to_I C2, whereC2denotes the cyclic group of order two.Ih

is isomorphic to the Coxeter groupH3(Humphreys, 1990) and

is made up of 120 elements. In this work, we focus on the icosahedral group I because it plays a central role in appli-cations in virology (Indelicato et al., 2011). However, our considerations apply equally to the larger group_Ih.

Levitov & Rhyner (1988) classified the Bravais lattices inR6

that are left invariant by _I: there are, up to equivalence, exactly three lattices, usually referred to as icosahedral Bravais lattices, namely the simple cubic (SC), body-centred cubic (BCC) and face-centred cubic (FCC). The point group of these lattices is the six-dimensional hyperoctahedral group,

denoted by B6, which is a subgroup of Oð6Þ and can be

represented in the standard basis ofR6_{as the set of all 66}

orthogonal and integral matrices. The subgroups ofB6which

are isomorphic to the icosahedral group constitute the integral representations of it; among them, the crystallographic ones are those which split, inGL_ð6;R_{Þ, into two three-dimensional}

irreducible representations of _I. Therefore, they carry two subspaces inR3which are invariant under the action of_I and can be used to model the quasiperiodic structures.

The embedding of the icosahedral group intoB6 has been

used extensively in the crystallographic literature. Katz (1989), Senechal (1995), Kramer & Zeidler (1989), Baake & Grimm (2013), among others, start from a six-dimensional crystal-lographic representation of_I to construct three-dimensional Penrose tilings and icosahedral quasicrystals. Kramer (1987) and Indelicato et al. (2011) also apply it to study structural transitions in quasicrystals. In particular, Kramer considers in B6a representation ofI and a representation of the

octahe-dral group_Owhich share a tetrahedral subgroup, and defines a continuous rotation (called Schur rotation) between cubic and icosahedral symmetry which preserves intermediate tetrahedral symmetry. Indelicato et al. define a transition between two icosahedral lattices as a continuous path connecting the two lattice bases keeping some symmetry preserved, described by a maximal subgroup of the icosahe-dral group. The rationale behind this approach is that the two corresponding lattice groups share a common subgroup. These two approaches are shown to be related (Indelicato et al., 2012), hence the idea is that it is possible to study the transi-tions between icosahedral quasicrystals by considering two distinct crystallographic representations of _I in B6 which

share a common subgroup.

These papers motivate the idea of studying in some detail the subgroup structure ofB6. In particular, we focus on the

subgroups isomorphic to the icosahedral group and its subgroups. Since the group is quite large (it has 26_6!elements),

the elements of B6 inGAPas a subgroup of the symmetric

groupS12and then find the classes of subgroups isomorphic to

the icosahedral group. Among them we isolate, using results 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_Gof_Iand 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 topology, hence describing the intersection and the subgroups shared by different representations.

The paper is organized as follows. After recalling, in_x2, the definitions of point group and lattice group, we define, in _x3, the crystallographic representations of the icosahedral group and the icosahedral lattices in six dimensions. We provide, following Kramer & Haase (1989), a method for the construction of the projection into three dimensions using tools from the representation theory of finite groups. In_x4 we classify, with the help of GAP, the crystallographic repre-sentations of _I. In _x5 we study their subgroup structure, introducing the concept of _G-graph, where _G is a subgroup of_I.

2. Lattices and noncrystallographic groups

Letb_i,i¼1;. . ._{;nbe a basis of}Rn_{, and letB}

2GL_ðn;R_Þbe

the matrix whose columns are the components of b_{i with}

respect to the canonical basis_fe_i;i_¼1;. . .;n_gofRn_{. A lattice}

inRn is aZ-free module of ranknwith basisB,i.e.

LðB_{Þ ¼}

x_¼P

n

i¼1

mibi: mi2Z

:

Any other lattice basis is given byBM, whereM₂GL_ðn;Z_Þ, the set of invertible matrices with integral entries (whose determinant is equal to1) (Artin, 1991).

The point group of a lattice_Lis given by all the orthogonal transformations that leave the lattice invariant (Pitteri & Zanzotto, 2002):

PðB_{Þ ¼ f}Q₂O_ðn_Þ: _9M2GLðn;Z_Þs:t:QB_¼BM_g:

We notice that, ifQ_{2 Pð}B_Þ, thenB1QB_¼M₂GL_ðn;Z_Þ.

In other words, the point group consists of all the orthogonal matrices which can be represented in the basisBas integral matrices. The set of all these matrices constitute the lattice group of the lattice:

_{ðBÞ ¼ fM2GLðn;}Z_Þ: _{9Q2 PðBÞs:t:M¼B}1

QB_g:

The lattice group provides an integral representation of the point group and these are relatedviathe equation

_{ðBÞ ¼B}1_PðBÞB;

and moreover the following hold (Pitteri & Zanzotto, 2002):

PðBM_{Þ ¼ Pð}B_Þ; _{ðBMÞ ¼M}1_{ðBÞM; M2GLðn;}Z_Þ:

We notice that a change of basis in the lattice leaves the point group invariant, whereas the corresponding lattice groups are conjugated inGL_ðn;Z_{Þ. Two lattices are}

inequi-valent if the corresponding lattice groups are not conjugated inGL_ðn;Z_{Þ(Pitteri & Zanzotto, 2002).}

As a consequence of the crystallographic restriction [see, for example, Baake & Grimm (2013)] five- and n-fold symmetries, wherenis a natural number greater than six, are forbidden in dimensions two and three, and therefore any group G containing elements of such orders cannot be the point group of a two- or three-dimensional lattice. We there-fore call these groups noncrystallographic. In particular, three-dimensional icosahedral lattices cannot exist. However, a noncrystallographic group leaves some lattices invariant in higher dimensions and the smallest such dimension is called the minimal embedding dimension. Following Levitov & Rhyner (1988), we introduce:

Definition 2.1. Let G be a noncrystallographic group. A crystallographic representation of G is a D-dimensional representation ofGsuch that:

(1) the charactersofare integers;

(2)is reducible and contains a two- or three-dimensional representation ofG.

We observe that the first condition implies thatGmust be the subgroup of the point group of aD-dimensional lattice. The second condition tells us that contains either a two-or three-dimensional invariant subspace E of RD_{, usually}

referred to as physical space (Levitov & Rhyner, 1988).

3. Six-dimensional icosahedral lattices

The icosahedral group_I is generated by two elements,g₂and g₃, such that g2

2¼g33¼ ðg2g3Þ 5

¼e, where e denotes the identity element. It has size 60 and it is isomorphic toA₅, the alternating group of order five (Artin, 1991). Its character table is given in Table 1.

From the character table we see that the (minimal) crys-tallographic representation of_Iis six-dimensional and is given by T1T2. Therefore, I leaves a lattice in R

6

[image:3.610.315.565.105.189.2]

invariant. Levitov & Rhyner (1988) proved that the three inequivalent Bravais lattices of this type, mentioned in theIntroductionand

Table 1

Character table of the icosahedral group.

referred to as icosahedral (Bravais) lattices, are given by, respectively:

LSC¼ fx¼ ðx1;. . .;x6Þ:xi2Zg;

LBCC¼ _x

¼1

2ðx1;. . .;x6Þ:xi2Z;xi¼xjmod2;8i;j¼1;. . .;6

;

LFCC ¼ _x

¼1

2ðx1;. . .;x6Þ:xi2Z;

P6

i¼1xi¼0 mod2

:

We note that a basis of the SC lattice is the canonical basis ofR6_{. Its point group is given by}

PSC¼ fQ2Oð6Þ:Q¼M2GLð6;ZÞg ¼Oð6Þ \GLð6;ZÞ ’Oð6;ZÞ;

ð1_Þ

which is the hyperoctahedral group in dimension six. In the following, we will denote this group by B6, following

Humphreys (1996). We point out that this notation comes from Lie theory: indeed,B6represents the root system of the

Lie algebra soð13Þ (Fulton & Harris, 1991). However, the

corresponding reflection group W_ðB_6Þ is isomorphic to the hyperoctahedral group in six dimensions (Humphreys, 1990). All three lattices have point groupB₆, whereas their lattice groups are different and, indeed, they are not conjugate in GL_ð6;Z_Þ(Levitov & Rhyner, 1988).

Let_Hbe a subgroup ofB6 isomorphic toI. Hprovides a

(faithful) integral and orthogonal representation of_I. More-over, if _{H ’}T1T2 in GLð6;RÞ, then H is also

crystal-lographic (in the sense of Definition 2.1). All of the other crystallographic representations are given by B1_HB, where B₂GL_ð6;R_{Þ 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_Hbe a crystallographic representation of the icosahe-dral group. _H splits into two three-dimensional irreducible representations (IRs),T1andT2, inGLð6;RÞ. This means that

there exists a matrixR₂GL_ð6;R_{Þsuch that}

H0:_¼R1_HR¼ T1 0

0 T2 !

: _ð2_Þ

The two IRs T1 and T2 leave two three-dimensional

subspaces invariant, which are usually referred to as the physical (or parallel) space Ek _{and the orthogonal spaceE}?

(Katz, 1989). In order to find the matrixR(which is not unique in general), we follow (Kramer & Haase, 1989) and use results from the representation theory of finite groups (for proofs and further results see, for example, Fulton & Harris, 1991). In particular, let :_{G!GLðn;FÞbe ann-dimensional} repre-sentation of a finite group G over a field F (F = R_, C_).

By Maschke’s theorem, splits, in GL_ðn;F_Þ, as m1 1. . .mr r, where i:G!GLðni;FÞ is an ni

-dimensional IR of G. Then the projection operator P_i:_Fn

!Fni is given by

P_i:_¼ ni

jG_j X

g2I

iðgÞ ðgÞ; ð3Þ

where

i denotes the complex conjugate of the character of the representation _i. This operator is such that its image

Im_ðPiÞis equal to anni-dimensional subspaceViofFn

invar-iant under i. In our case, we have two projection operators,

P_i:R6_!R3, i= 1, 2, corresponding to the IRsT₁ andT₂, respectively. We assume the image ofP₁, Im_ðP_1Þ, to be equal to Ek_{, and ImðP}

2Þ ¼E?. Iffej;j¼1;. . .;6gis the canonical basis

of R6, then a basis of Ek _{(respectively E}?_{) can be found}

considering the set _fee^

j:¼Piej;j¼1;. . .;6g for i¼1

(respectively i_¼2) and then extracting a basis _Bi from it. Since dimEk_{= dimE}?_{¼3, we obtainB}

i¼ fee^i;1;ee^i;2;ee^i;3g, fori

= 1, 2. The matrixRcan be thus written as

R_¼

^

e

e_1;1;ee^_1;2;ee^_1;3 |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}

basis ofEk

;ee^_2;1;ee^_2;2;ee^_2;3 |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}

basis ofE?

: _ð4_Þ

Denoting byk_and?_{the 36 matrices which represent}

P₁ andP₂ in the bases _B1 and_B2, respectively, we have, by linear algebra

R1_¼ k

?

!

: _ð5_Þ

SinceR1

H ¼ H0R1_{[cf.equation (2)], we obtain}

k_ðHðg_Þv_{Þ ¼}T_1ðk_ðv_ÞÞ; ?_ðHðg_Þv_{Þ ¼}T_2ð?_ðv_ÞÞ; _ð6_Þ

for allg_{2 I}andv₂R6. In particular, the following diagram commutes

The set_ðH; k_{Þis the starting point for the construction of}

quasicrystalsviathe cut-and-project method (Senechal, 1995; Indelicatoet al., 2012).

4. Crystallographic representations of_I

From the previous section it follows that the six-dimensional hyperoctahedral groupB6 contains all the (minimal)

ortho-gonal crystallographic representations of the icosahedral group. In this section we classify them, with the help of the computer software programmeGAP(The GAP Group, 2013).

4.1. Representations of the hyperoctahedral groupB₆ Permutation representations of the n-dimensional hyper-octahedral groupBnin terms of elements ofS2n, the symmetric

group of order 2n, have been described by Baake (1984). In this subsection, we review these results because they allow us to generateB₆inGAPand further study its subgroup struc-ture.

It follows from equation (1) that B6 consists of all the

orthogonal integral matrices. A matrixA_{¼ ð}a_ijÞof this kind must satisfy AAT _¼I

6, the identity matrix of order six, and

permutation matrices. It is straightforward to see that anyA₂B6can be written in the form

NQ, where Qis a 66 permutation matrix andNis a diagonal matrix with each diagonal entry being either 1 or 1. We can thus associate with each matrix in B6a pairða; Þ,

wherea₂Z6₂is a vector given by the diagonal elements ofN, and₂S6is the permutation

associated with Q. The set of all these pairs constitutes a group (called the wreath product of Z_{2 and S6, and denoted by} Z_2oS6;

Humphreys, 1996) with the multiplication rule given by

ða_{; Þð}b_{; Þ}:_{¼ ð}a

þ2b; Þ;

where_þ2denotes addition modulo 2 and

ða_Þ

k:¼aðkÞ; a¼ ða1;. . .;a6Þ:

Z_{2oS6andB6are isomorphic, an isomorphism}

Tbeing the following:

½T_ða_{; Þ}

ij:¼ ð1Þ ai

ðiÞ;j: ð8Þ

It immediately follows that |B6| = 266! = 46 080. A set of

generators is given by

:_{¼ ð}0_;ð1;2ÞÞ; :_{¼ ð}0_{;ð1;2;3;4;5;6ÞÞ;} :_{¼ ðð0;0;0;0;0;1Þ;id}

S6Þ; ð9Þ

which satisfy the relations

2_¼2_¼ 6_{¼ ð}0_;id S6Þ:

Finally, the function’:Z_{2oS6!S12defined by}

’_ða; _Þðk_Þ:_¼ (

_ðk_{Þ þ}6ak if 1 k 6 _ðk6_{Þ þ}6_ð1a_k6Þ if 7 k 12

ð10_Þ is injective and maps any element ofZ_2oS₆into a permutation of S₁₂, and provides a faithful permutation representation of B₆ as a subgroup of S₁₂. Combining equation (8) with the inverse of equation (10) we get the function

:_¼T1_:S

12 !B6; ð11Þ

which can be used to map a permutation into an element ofB6.

4.2. Classification

In this subsection we classify the orthogonal crystal-lographic representations of the icosahedral group. We start by recalling a standard way to construct such a representation, following Zappaet al.(2013). We consider a regular icosahe-dron and we label each vertex by a number from one to 12, so that the vertex opposite to vertexi is labelled byi_þ6 (see Fig. 1). This labelling induces a permutation representation

:_{I !S12 given by}

_ðg_{2Þ ¼ ð}1;6_Þð2;5_Þð3;9_Þð4;10_Þð7;12_Þð8;11_Þ; _ðg_{3Þ ¼ ð}1;5;6_Þð2;9;4_Þð7;11;12_Þð3;10;8_Þ:

Using equation (11) we obtain a representation _II^ :_{I !B} 6

given by

^

IIðg_{2Þ ¼}

0 0 0 0 0 1 0 0 0 0 1 0 0 01 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 0

0

B B B B B B @

1

C C C C C C A

; _IIð^ g_{3Þ ¼}

0 0 0 0 0 1 0 0 0 1 0 0 01 0 0 0 0 0 01 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0

0

B B B B B B @

1

C C C C C C A :

ð12Þ

We see that_II^ðg₂Þ ¼ 2 and_II^ðg₃Þ ¼0, so that, by looking

at the character table of_I, we have

^

II ¼T1 þT2;

which implies, using Maschke’s theorem (Fulton & Harris, 1991), that_{II ’}^ T1T2inGLð6;RÞ. Therefore, the subgroup

^

II ofB6is a crystallographic representation ofI.

Before we continue, we recall the following (Humphreys, 1996):

Definition 4.1. Let H be a subgroup of a group G. The conjugacy class ofHinGis the set

CGðHÞ:¼ fgHg 1_:g

2G_g:

In order to find all the other crystallographic representa-tions, we use the following scheme:

(a) we generateB6as a subgroup ofS12using equations (9)

and (10);

(b) we list all the conjugacy classes of the subgroups ofB₆ and find a representative for each class;

(c) we isolate the classes whose representatives have order 60;

(d) we check if these representatives are isomorphic to_I; (e) we map these subgroups of S12 intoB6using equation

(11) and isolate the crystallographic ones by checking the characters; denoting bySthe representative, we decompose_S

[image:5.610.253.564.68.239.2]

as

Figure 1

S¼m1Aþm2T1þm3T2þm4Gþm5H;

m_i2N_{; i¼1;}. . ._;5:

Note thatSis crystallographic if and only ifm2¼m3¼1

andm1¼m4¼m5¼0.

We implemented steps (1)–(4) inGAP(see Appendix C). There are three conjugacy classes of subgroups isomorphic to I inB₆. Denoting byS_{i¼ h}g_2;i;g_3;iithe representatives of the classes returned byGAP, we have, using equation (11),

S₁ðg2;1Þ ¼2; S₁ðg3;1Þ ¼3)S₁ ¼2AþG)S1’2AG; S₂ðg2;2Þ ¼ 2; S₂ðg3;2Þ ¼0)S₂ ¼T₁þT₂)S2’T1T2; _S

3ðg2;3Þ ¼2; S3ðg3;3Þ ¼0)S3 ¼AþH)S3’AH:

Since 2A is decomposable into two one-dimensional representations, it is not strictly speaking two dimensional in the sense of Definition 2.1, and as a consequence, only the second class contains the crystallographic representations of I. A computation inGAPshows that its size is 192. We thus have the following:

Proposition 4.1. The crystallographic representations of_Iin B6form a unique conjugacy class in the set of all the classes of

subgroups ofB6, and its size is equal to 192.

We briefly point out that the other two classes of subgroups isomorphic to _I inB6 have an interesting algebraic

intepre-tation. First of all, we observe that B6is an extension of S6,

since according to Humphreys (1996):

B6=Z 6

2’ ðZ2oS6Þ=Z 6 2’S6:

Following Janusz & Rotman (1982), it is possible to embed the symmetric group S5 into S6 in two different ways. The

canonical embedding is achieved by fixing a point in_f1;. . ._;6g and permuting the other five, whereas the other embedding is by means of the so-called ‘exotic map’’:_{S5!S6, which acts} on the six 5-Sylow subgroups ofS5by conjugation. Recalling

that the icosahedral group is isomorphic to the alternating groupA₅, which is a normal subgroup ofS₅, then the canonical embedding corresponds to the representation 2AGinB6,

while the exotic one corresponds to the representationAH. In what follows, we will consider the subgroup_II^ previously defined as a representative of the class of the crystallographic representations of_I, and denote this class by_CB6ðIIÞ^ .

Recalling that two representationsDð1Þ andDð2Þof a group Gare said to be equivalent if there are relatedviaa similarity transformation,i.e.there exists an invertible matrixSsuch that

Dð1Þ_¼SDð2ÞS1;

then an immediate consequence of Proposition 4.1 is the following:

Corollary 4.1. Let _H1 andH2 be two orthogonal

crystal-lographic representations of_I. Then_H1andH2are

equiva-lent inB6.

We observe that the determinant of the generators of_II^ in equation (12) is equal to one, so that _{II 2}^ Bþ6 :¼

fA₂B6:detA¼1g. Proposition 4.1 implies that all the

crystallographic representations belong to Bþ6. The

remark-able fact is that they split into two different classes inBþ6. To

see this, we first need to generateBþ6. In particular, withGAP

we isolate the subgroups of index two inB₆, which are normal inB₆, and then, using equation (11), we find the one whose generators have determinant equal to one. In particular, we have

Bþ6 ¼ hð1;2;6;4;3Þð7;8;12;10;9Þ;ð5;11Þð6;12Þ;

ð1;2;6;5;3_Þð7;8;12;11;9_Þ;_ð5;12;11;6_Þi:

We can then apply the same procedure to find the crystal-lographic representations of_I, and see that they split into two classes, each one of size 96. Again we can choose _II^ as a representative for one of these classes; a representative_KK^ for the other one is given by

^ K K ¼

*

0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0

B B B B B B @

1

C C C C C C A

;

0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0

B B B B B B @

1

C C C C C C A +

:

ð13_Þ We note that in the more general case of _Ih, we can construct the crystallographic representations of _Ih starting from the crystallographic representations of_I. First of all, we recall that_Ih¼IC2, whereC2is the cyclic group of order

two. Let_Hbe a crystallographic representation of_IinB6, and

let _{¼ f}1;1_g be a one-dimensional representation of C2.

Then the representation_HH^ given by

^ H

H:_{¼ H ;}

where denotes the tensor product of matrices, is a repre-sentation of_IhinB₆and it is crystallographic in the sense of Definition 2.1 (Fulton & Harris, 1991).

4.3. Projection into the three-dimensional space

We study in detail the projection into the physical spaceEk using the methods described in_x3.1.

Let_II^ be the crystallographic representation of_I given in equation (12). Using equation (3) with n_i¼3 and _jG_{j ¼} jIj ¼60 we obtain the following projection operators

P_1¼ 1 2pffiffiffi5

ffiffiffi 5 p

1 1 1 1 1

1 pffiffiffi5 1 1 1 1

1 1 pffiffiffi5 1 1 1

1 1 1 pffiffiffi5 1 1

1 1 1 1 pffiffiffi5 1 1 1 1 1 1 pffiffiffi5 0

B B B B B B @

1

C C C C C C A

;

P_2¼ 1 2pffiffiffi5

ffiffiffi 5 p

1 1 1 1 1

1 pffiffiffi5 1 1 1 1

1 1 pffiffiffi5 1 1 1 1 1 1 pffiffiffi5 1 1

1 1 1 1 pffiffiffi5 1

1 1 1 1 1 pffiffiffi5

0

B B B B B B @

1

C C C C C C A

The rank of these operators is equal to three. We choose as a basis of Ek and E? the following linear combination of the columns c_{i;j of the projection operators Pi, for i = 1, 2 and}

j_¼1;. . ._;6:

c₁ ;1þc1;5

2 ; c₁

;2c1;4 2 ;

c₁ ;3þc1;6

[image:7.610.46.300.91.235.2] [image:7.610.367.516.206.290.2] [image:7.610.45.296.299.396.2] [image:7.610.48.296.478.513.2]

2

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} basis ofEk

; c2;1c2;5

2 ; c₂

;2þc2;4 2 ;

c₂ ;3c2;6

2

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} basis ofE?

!

:

With a suitable rescaling, we obtain the matrixRgiven by

R_¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 2_ð2_þÞ

p

1 0 0 1

0 11 0

1 0 0 1

0 1 1 0

1 0 0 1 1 0 0 1

0

B B B B B B @

1

C C C C C C A

:

The matrixRis orthogonal and reduces_II^as in equation (2). In Table 2 we give the explicit forms of the reduced repre-sentation. The matrix representation inEk _{of P}

1 is given by

[see equation (5)]

k_¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 2_ð2_þÞ

p

01 0 1 1 0 1 0

0 1 1 0

0

@

1

A:

The orbit_fT1ðkðejÞÞg, wherefej;j¼1;. . .;6g is the

cano-nical basis of R6_{, represents a regular icosahedron in three}

dimensions centred at the origin (Senechal, 1995; Katz, 1989; Indelicatoet al., 2011).

Let_Kbe another crystallographic representation of_IinB6.

By Proposition 4.1,_K and_II^ are conjugated inB6. Consider

M₂B6such thatMII^M1¼ Kand letS¼MR. We have

S1KS¼ ðMRÞ1KðMRÞ ¼R1M1KMR¼R1II^R¼T1T2:

Therefore it is possible, with a suitable choice of the reducing matrices, to project all the crystallographic representations of I inB6in the same physical space.

5. Subgroup structure

The nontrivial subgroups of_I are listed in Table 3, together with their generators (Hoyle, 2004). Note that_T,_D10 andD6

are maximal subgroups of _I, and that _D4, C5 and C3 are

normal subgroups of_T,_D10andD6, respectively (Humphreys,

1996; Artin, 1991). The permutation representations of the generators inS₁₂are given in Table 4 (see also Fig. 1).

Since_Iis a small group, its subgroup structure can be easily obtained in GAP by computing explicitly all its conjugacy classes of subgroups. In particular, there are seven classes of nontrivial subgroups in _I: any subgroup H of _I has the property that, ifKis another subgroup of_I isomorphic toH, thenHandKare conjugate in_I(this property is referred to as the ‘friendliness’ of the subgroupH; Soicher, 2006). In other words, denoting by n_G the number of subgroups of _I isomorphic to_G,i.e.

n_G:_{¼ jfH<I} :_{H’ Ggj; ð14Þ}

we have (cf.Definition 4.1)

n_{G¼ jCIðGÞj}:

In Table 5 we list the size of each class of subgroups in_I. Geometrically, different copies ofC2,C3andC5correspond to

the different two-, three- and fivefold axes of the icosahedron, respectively. In particular, different copies of_D10stabilize one

Table 2

Explicit forms of the IRsT₁andT₂withII ’^ T₁T₂.

Generator IrrepT₁ IrrepT₂

g₂ 1

2

1 1

1 1

1 1

0

@

1

A 1

2

1 1

1 1

1 1 0

@

1

A

g₃ 1

2

1 1

1 1

1 1

0

@

1

A 1

2

1 1

1 1

1 1

0

@

1

A

Table 3

Nontrivial subgroups of the icosahedral group.

T stands for the tetrahedral group,D2nfor the dihedral group of size 2n, and

Cnfor the cyclic group of sizen.

Subgroup Generators Relations Size

T g₂;g_3d g2

2¼g33d¼ ðg2g3dÞ

3_{¼e 12}

D10 g2d;g5d g22d¼g55d¼ ðg5dg2dÞ

2

¼e 10

D6 g2d;g3 g22d¼g33¼ ðg3g2dÞ2¼e 6

C₅ g_5d g5

5d¼e 5

D4 g2d;g2 g22d¼g22¼ ðg2g2dÞ

2

¼e 4

C₃ g₃ g3

3¼e 3

C₂ g₂ g2

2¼e 2

Table 4

Permutation representations of the generators of the subgroups of the icosahedral group.

ðg₂Þ ¼ ð1;6Þð2;5Þð3;9Þð4;10Þð7;12Þð8;11Þ ðg_2dÞ ¼ ð1;12Þð2;8Þð3;4Þð5;11Þð6;7Þð9;10Þ _ðg3Þ ¼ ð1;5;6Þð2;9;4Þð7;11;12Þð3;10;8Þ

ðg₃dÞ ¼ ð1;10;2Þð3;5;12Þð4;8;7Þð6;9;11Þ _ðg_{5Þ ¼ ð}1;2;3;4;5Þð7;8;9;10;11Þ ðg₅dÞ ¼ ð1;10;11;3;6Þð4;5;9;12;7Þ

Table 5

Sizes of the classes of subgroups of the icosahedral group inIandB6.

Subgroup jCIðGÞj jCB6ðKGÞj

T 5 480

D10 6 576

D6 10 960

D4 5 120

C₅ 6 576

C₃ 10 320

of the six fivefold axes of the icosahedron, and each copy of_D6

stabilizes one of the ten threefold axes. Moreover, it is possible to inscribe five tetrahedra into a dodecahedron, and each different copy of the tetrahedral group in _I stabilizes one of these tetrahedra.

5.1. Subgroups of the crystallographic representations of_I Let _G be a subgroup of _I. The function (11) provides a representation of_GinB6, denoted byKG, which is a subgroup

of_II^. Let us denote by_CB6ðKGÞthe conjugacy class ofKGinB6.

The next lemma shows that this class contains all the subgroups of the crystallographic representations of_I inB₆.

Lemma 5.1. Let _Hi2 CB6ð

^

IIÞ be a crystallographic repre-sentation of _I in B₆ and let _{Ki Hi} be a subgroup of _Hi isomorphic to_G. Then_{Ki2 CB}

6ðKGÞ.

Proof. Since _Hi2CB6ð

^

IIÞ, there exists g₂B6 such that

g_Hig1¼II^, and therefore gKig1¼ K0 is a subgroup of II^

isomorphic to _G. Since all these subgroups are conjugate in ^

II [they are ‘friendly’ in the sense intended by Soicher (2006)], there exists h_2II^ such that h_K0h1

¼ KG. Thus

ðhg_ÞKiðhg_Þ1_{¼ KG}, implying that_{Ki2 CB}

6ðKGÞ. &

We next show that every element of_CB

6ðKGÞis a subgroup

of a crystallographic representation of_I.

Lemma 5.2. Let _Ki2 CB6ðKGÞ. There exists Hi2 CB6ð

^ IIÞ such that_Kiis a subgroup of_Hi.

Proof. Since _{Ki2 CB6ðKGÞ}, there exists g₂B6 such that

g_Kig1¼ KG. We defineHi:¼g1II^g. It can be seen

imme-diately that_Kiis a subgroup ofHi. &

As a consequence of these lemmata, _CB6ðKGÞ contains all the subgroups of B6 which are isomorphic to G and are

subgroups of a crystallographic representation of _I. The explicit forms of _KG are given in Appendix B. We point out that it is possible to find subgroups of B₆ isomorphic to a subgroup _G of _I which are not subgroups of any crystal-lographic representation of _I. For example, the following subgroup

^ TT ¼

*

1 0 0 0 0 0 01 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0

B B B B B B @

1

C C C C C C A

;

0 0 1 0 0 0 1 0 0 0 0 0 01 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0

B B B B B B @

1

C C C C C C A +

is isomorphic to the tetrahedral group _T; a computation in GAPshows that it is not a subgroup of any elements in_CB6ðIIÞ^ . Indeed, the two classes of subgroups,_CB6ðKTÞand_{CB6ðTT Þ}^ , are disjoint.

UsingGAP, we compute the size of each_CB6ðKGÞ(see Table 5). We observe that_jCB6ðKGÞj<_jCB6ðIIÞj ^ n_G. This implies that

crystallographic representations of_Imay share subgroups. In order to describe more precisely the subgroup structure of CB6ð

^

IIÞwe will use some basic results from graph theory and their spectra, which we are going to recall in the next section.

5.2. Some basic results of graph theory and their spectra In this section we recall, without proofs, some concepts and results from graph theory and spectral graph theory. Proofs and further results can be found, for example, in Foulds (1992) and Cvetkovicet al.(1995).

Let G be a graph with vertex set V_{¼ f}v₁;. . ._;v ng. The

number of edges incident with a vertexvis called the degree of v. If all vertices have the same degreed, then the graph is called regular of degreed. A walk of lengthlis a sequence ofl consecutive edges and it is called a path if they are all distinct. 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 verticespandqare connected if there exists a path containing pandq. The connected component of a vertexvis the set of all vertices connected tov.

The adjacency matrixAofGis thennmatrixA_{¼ ð}a_ijÞ whose entriesa_ijare equal to one if the vertexv_iis adjacent to the vertexvj, and zero otherwise. It can be seen immediately

from its definition thatAis symmetric andaii¼0 for alli, so

that Tr_ðA_{Þ ¼}0. It follows thatAis diagonalisable and all its eigenvalues are real. The spectrum of the graph is the set of all the eigenvalues of its adjacency matrixA, usually denoted by

_ðA_Þ.

Theorem 5.1. LetAbe the adjacency matrix of a graphG with vertex set V_{¼ f}v₁;. . ._;v

ng. Let Nkði;jÞ denote the

number of walks of lengthkstarting at vertexv_iand finishing at vertexv_j. We have

N_kði;j_{Þ ¼}Ak_ij:

We recall that the spectral radius of a matrixAis defined by

_ðA_Þ:_¼maxfjj:_{2ðAÞg. IfAis a non-negative matrix,i.e.} if all its entries are non-negative, then_ðA_{Þ 2ð}A_Þ(Horn & Johnson, 1985). Since the adjacency matrix of a graph is non-negative,_{jj ð}A_Þ:_{¼r, where2ðAÞandris the largest} eigenvalue.ris called the index of the graphG.

Theorem 5.2. Let1;. . .; nbe the spectrum of a graphG,

and letrdenote its index. ThenGis regular of degreerif and only if

1 n

Xn

i¼1 2_{i ¼}r:

Moreover, ifGis regular the multiplicity of its index is equal to the number of its connected components.

5.3. Applications to the subgroup structure

Let_Gbe a subgroup of_I. In the following we represent the subgroup structure of the class of crystallographic repre-sentations of _I in B6, CB6ð

^

H1;H22 CB6ð

^

IIÞare adjacent to each other (i.e.connected by an edge) in the graph if there exists P_{2 C}B6ðKGÞ such that

P_{¼ H1\ H2}. We can therefore consider the graph G_{¼ ðCB}

6ð

^

IIÞ;E_Þ, where an edgee₂Eis of the form_ðH1;_H2Þ. We call this graph_G-graph.

Using GAP, we compute the adjacency matrices of the G-graphs. The algorithms used are shown in Appendix C. The spectra of the _G-graphs are given in Table 6. We first of all notice that the adjacency matrix of the C5-graph is the null

matrix, implying that there are no two representations sharing precisely a subgroup isomorphic to C5, i.e. not a subgroup

containingC5. We point out that, since the adjacency matrix of

the _D10-graph is not the null one, then there exist

crystal-lographic representations, say_Hi and_Hj, sharing a maximal subgroup isomorphic to_D10. SinceC5is a (normal) subgroup

of_D10, thenHiandHjdo share aC5subgroup, but also aC2

subgroup. In other words, if two representations share a fivefold axis, then necessarily they also share a twofold axis.

A straightforward calculation based on Theorem 5.2 leads to the following

Proposition 5.1. Let _G be a subgroup of _I. Then the corresponding_G-graph is regular.

In particular, the degreed_Gof each_G-graph is equal to the largest eigenvalue of the corresponding spectrum. As a consequence we have the following:

Proposition 5.2. Let_Hbe a crystallographic representation of _I in B6. Then there are exactly dG representations

Kj2 CB6ð

^

IIÞsuch that

H \ Kj¼Pj;9Pj2 CðKGÞ;j¼1;. . .;dG:

In particular, we haved_G = 5, 6, 10, 0, 30, 20, 60 and 60 for G ¼ T;_D10;D6,C5;D4;C3;C2andfeg, respectively.

In particular, this means that for any crystallographic representation of_I there are precisely d_G other such repre-sentations which share a subgroup isomorphic to_G. In other words, we can associate to the class _CB6ð

^

IIÞ the ‘subgroup matrix’Swhose entries are defined by

Sij¼ jHi\ Hjj; i;j¼1;. . .;192:

The matrixSis symmetric andS_ii¼60, for all i, since the order of_I is 60. It follows from Proposition 5.2 that each row of Scontainsd_Gentries equal to_jGj. Moreover, a rearrange-ment of the columns ofSshows 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 D4-subgroup. This implies that the corresponding subgraph of

the _D4-graph is a complete graph, i.e. every two distinct vertices are connected by an edge. From a geometric point of view, these 16 representations correspond to ‘six-dimensional icosahedra’. This ensemble of 16 such icosahedra embedded into a dimensional hypercube can be viewed as a six-dimensional analogue of the three-six-dimensional ensemble of five tetrahedra inscribed into a dodecahedron, sharing pair-wise aC3-subgroup.

We notice that, using Theorem 5.2, not all the graphs are connected. In particular, the_D10- and theD6-graphs are made

up of six connected components, whereas theC₃- and theC₂ -graphs consist of two connected components. WithGAP, we implemented a breadth-first search algorithm (Foulds, 1992), which starts from a vertexiand then ‘scans’ for all the vertices connected to it, which allows us to find the connected components of a given _G-graph (see Appendix C). We find that each connected component of the_D10- andD6-graphs is

made up of 32 vertices, while for theC3- andC2-graphs each

component consists of 96 vertices. For all other subgroups, the corresponding _G-graph is connected and the connected component contains trivially 192 vertices.

We now consider in more detail the case when _G is a maximal subgroup of_I. Let_{H 2 C}B6ð

^

IIÞand let us consider its vertex star in the corresponding_G-graph,i.e.

V_ðHÞ:_{¼ fK 2 C} B6ð

^

IIÞ:_{Kis adjacent toHg: ð15Þ}

A comparison of Tables 5 and 6 shows thatd_G¼n_G[i.e.the number of subgroups isomorphic to_Gin_I,cf.equation (14)] and therefore, since the graph is regular,_jV_{ðHÞj ¼}d_G¼n_G. This suggests that there is a one-to-one correspondence between elements of the vertex star of_Hand subgroups of_H isomorphic to_G; in other words, if we fix any subgroupPof_H isomorphic to _G, then P ‘connects’ _H with exactly another representation_K. We thus have the following:

Proposition 5.3. Let_Gbe a maximal subgroup of_I. Then for every P_{2 CB6ðKGÞ} there exist exactly two crystallographic representations of_I,_H1;H22 CB6ð

^

IIÞ, such thatP_{¼ H}1\ H2.

In order to prove it, we first need the following lemma:

Lemma 5.3. Let_Gbe a maximal subgroup of_I. Then the corresponding_G-graph is triangle-free,i.e.it has no circuits of length three.

Proof. LetA_G be the adjacency matrix of the_G-graph. By Theorem 5.1, its third power A3_G determines the number of walks of length three, and in particular its diagonal entries, ðA3

GÞii, fori¼1;. . .;192, correspond to the number of

trian-gular circuits starting and ending in vertex i. A direct computation shows that_ðA3

GÞii¼0, for alli, thus implying the

non-existence of triangular circuits in the graph. &

Proof of Proposition 5.3. IfP_{2 CB}

6ðKGÞ, then, using Lemma

5.2, there exists_{H12 CB}

6ð

^

IIÞsuch thatPis a subgroup of_H1. Let us consider the vertex starV_ðH1Þ. We have_jV_{ðH1Þj ¼}d_G; we call its elements_H2;. . .;Hd_Gþ1. Let us suppose thatP is

not a subgroup of any_Hj, forj_¼2;. . ._;d

Gþ1. This implies

thatP does not connect _H1 with any of theseHj. However,

since_H1has exactlynG different subgroups isomorphic toG,

then at least two vertices in the vertex star, say_H2andH3, are

Q_{¼ H}1\ H2;Q¼ H1\ H3)Q¼ H2\ H3:

This implies that_H1,H2andH3form a triangular circuit in the

graph, which is a contradiction due to Lemma 5.3, hence the

result is proved. &

It is noteworthy that the situation inBþ₆ is different. If we denote by X1 and X2 the two disjoint classes of

crystal-lographic representations of _I inBþ6 [cf. equation (13)], we

can build, in the same way as described before, the_G-graphs forX1andX2, forG ¼ T;D10 andD6. The result is that the

adjacency matrices of all these six graphs are the null matrix of dimension 96. This implies that these graphs have no edges, and so the representations in each class do not share any maximal subgroup of _I. As a consequence, we have the following:

Proposition 5.4. Let_H;_{K 2 C}B6ð

^

IIÞbe two crystallographic representations of_I, andP_{¼ H \ K},P_{2 C}B6ðKGÞ, whereGis

a maximal subgroup of_I. Then_Hand_Kare not conjugated in Bþ₆. In other words, the elements ofB₆which conjugate_Hwith Kare matrices with determinant equal to1.

We conclude by showing a computational method which combines the result of Propositions 4.1 and 5.2. We first recall the following:

Definition 5.1. Let H be a subgroup of a group G. The normaliser ofHinGis given by

N_GðH_Þ:_{¼ fg2G}:_gHg1

¼H_g:

Corollary 5.1. Let_Hand_Kbe two crystallographic repre-sentations of_IinB6andP2 CðKGÞsuch thatP¼ H \ K. Let

A_H;K¼ fM2B6:MHM 1

¼ Kg

be the set of all the elements ofB6which conjugateHwithK

and letNB6ðPÞbe the normaliser ofPinB6. We have

A_H;K\N_B6ðH_{Þ 6¼ ;}:

In other words, it is possible to find a nontrivial element M₂B6in the normaliser ofPinB6which conjugatesHwith

K.

Proof. Let us suppose A_H;K\N_B

6ðHÞ ¼ ;. Then

MPM1_{6¼P, for all M2A}

H;K. This implies, since

M_HM1_{¼ K, that P is not a subgroup of K, which is a}

contradiction. &

We give now an explicit example. We consider the repre-sentation _II^ as in equation (12), and its subgroup _KD10 (the explicit form is given in Appendix B). WithGAP, we find the other representation _H02 CðIIÞ^ such thatKD10 ¼

^

II \ H0. Its

explicit form is given by

H0¼ *

0 0 0 01 0 0 0 0 1 0 0 0 01 0 0 0 0 1 0 0 0 0

1 0 0 0 0 0

0 0 0 0 0 1 0

B B B B B B @

1

C C C C C C A

;

0 0 0 0 1 0 0 01 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 01 0 0 0 0 0

B B B B B B @

1

C C C C C C A +

:

A direct computation shows that the matrix

M_¼

1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0

B B B B B B @

1

C C C C C C A

belongs to N_B6ðKD10Þ and conjugate _II^ with _H0. Note that

detM_¼ 1.

6. Conclusions

In this work we explored the subgroup structure of the hyperoctahedral group in six dimensions. In particular we found the class of the crystallographic representations of the icosahedral group, whose size is 192. Any such representation, together with its corresponding projection operatork_{, can be}

chosen to construct icosahedral quasicrystalsviathe cut-and-project method. We then studied in detail the subgroup structure of this class. For this, we proposed a method based on spectral graph theory and introduced the concept of_G-graph, for a subgroup_Gof the icosahedral group. This allowed us to study the intersection and the subgroups shared by different representations. We have shown that, if we fix any

repre-Table 6

Spectra of theG-graphs forGa nontrivial subgroup ofIandG ¼ fe_g, the trivial subgroup consisting of only the identity elemente.

The numbers highlighted are the indices of the graphs, and correspond to their degreesd_G.

T-graph D10-graph D6-graph C5-graph

Eig. Mult. Eig. Mult. Eig. Mult. Eig. Mult.

5 1 6 6 10 6 0 192

3 45 2 90 2 90

3 45 2 90 2 90

1 50 6 6 10 6

1 50

5 1

D4-graph C3-graph C2-graph feg-graph

Eig. Mult. Eig. Mult. Eig. Mult. Eig. Mult.

30 1 20 2 60 2 60 1

18 5 4 90 4 90 12 5

12 5 4 100 4 90 4 90

6 15 12 10 4 90

2 45 12 5

0 31 60 1

2 30

4 45

[image:10.610.313.567.117.557.2]

sentation_Hin the class and a maximal subgroupPof_H, then there exists exactly one other representation _K in the class such that P_{¼ H \ K}. As explained in theIntroduction, this can be used to describe transitions which keep intermediate symmetry encoded byP. In particular, this result implies in this context that a transition from a structure arising from_Hvia projection will result in a structure obtainable for _K via projection if the transition has intermediate symmetry described byP. Therefore, this setting is the starting point to analyse structural transitions between icosahedral quasicrys-tals, following the methods proposed in Kramer (1987), Katz (1989) and Indelicatoet 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 structures that can be modelledvialattices 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 Indelicatoet al. (2011), such structural transitions can be modelled using projections of six-dimensional icosahedral lattices and their 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.

APPENDIXA

In order to render this paper self-contained, we provide the character tables of the subgroups of the icosahedral group, following Artin (1991), Fulton & Harris (1991) and Jones (1990).

Tetrahedral group_T [!_¼exp_ð2i=3_Þ]:

Dihedral group_D10:

Dihedral group_D6(isomorphic to the symmetric groupS3):

Cyclic groupC5[¼expð2i=5Þ]:

Dihedral groupD4(the Klein Four Group):

Cyclic groupC3[!¼expð2i=3Þ]:

Cyclic groupC2:

APPENDIXB

Here we show the explicit forms of_KG, the representations inB6 of the subgroups ofI, together with their

decomposi-tions inGL_ð6;R_Þ.

KT ¼

*

0 0 0 0 0 1 0 0 0 0 1 0 0 01 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 0

0

B B B B B B @

1

C C C C C C A ;

0 1 0 0 0 0 0 0 01 0 0 0 0 0 0 0 1

1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 01 0

0

B B B B B B @

1

C C C C C C A +

;

KD10¼

*

0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 01 0

1 0 0 0 0 0

0

B B B B B B @

1

C C C C C C A ;

0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 01 0

1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0

0

B B B B B B @

1

C C C C C C A +

;

KD6¼

*

0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 01 0

1 0 0 0 0 0

0

B B B B B B @

1

C C C C C C A ;

0 0 0 0 0 1 0 0 0 1 0 0 01 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0

0

B B B B B B @

1

C C C C C C A +

KC5 ¼

*

0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 01 0

1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0

0

B B B B B B @

1

C C C C C C A +

;

KD4 ¼

*

0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 01 0

1 0 0 0 0 0

0

B B B B B B @

1

C C C C C C A ;

0 0 0 0 0 1 0 0 0 0 1 0 0 01 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 0

0

B B B B B B @

1

C C C C C C A +

:

KC3 ¼

*

0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0

0

B B B B B B @

1

C C C C C C A +

;

KC2¼

*

0 0 0 0 0 1 0 0 0 0 1 0 0 01 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 0

0

B B B B B B @

1

C C C C C C A +

:

KT ’2T;KD10’2A2E1E2; KD6’2A22E;KC5’2AE1E2;

KD4’2B12B22B3;KC3’2A2E;KC2’2A4B:

APPENDIXC

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 1–4 used to prove Proposition 4.1. In theGAPcomputation, the class_CB6ðIIÞ^ is indicated as CB6s60. Its size is 192.

Algorithm 2 (Fig. 3): Computation of the vertex star of a given vertexiin the_G-graphs. In the following,Hstands for the class_CB6ðIIÞ^ of the crystallographic representations of_I, i_{2 f}1;. . ._{;192gdenotes a vertex in theG-graph corresponding} to the representationH_½iandnstands 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 vertexiof the_G-graph.

Algorithm 5 (Fig. 6): Computation of all connected components of a_G-graph.

Figure 2

Algorithm 1.

Figure 3

Algorithm 2.

Figure 4

[image:12.610.48.295.69.373.2]

We would like to thank Silvia Steila, Pierre-Philippe Dechant, Paolo Cermelli and Giuliana Indelicato for useful discussions, and Paolo Barbero and Alessandro Marchino for technical help. ECD thanks the Leverhulme Trust for an Early Career Fellowship (ECF-2013-019) and the EPSRC for funding (EP/K02828671).

References

Artin, M. (1991).Algebra.New York: Prentice Hall. Baake, M. (1984).J. Math. Phys.25_{, 3171–3182.}

Baake, M. & Grimm, U. (2013).Aperiodic Order, Vol. 1. Cambridge University Press.

Cvetkovic, D., Doob, M. & Sachs, H. (1995). Spectra of Graphs. Heidelberg, Leipzig: Johann Ambrosius Barth.

Foulds, L. (1992).Graph Theory Applications.New York: Springer-Verlag.

Fulton, W. & Harris, J. (1991).Representation Theory: A First Course. Springer-Verlag.

Horn, R. & Johnson, C. (1985). Matrix Analysis. Cambridge University Press.

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.}

Indelicato, G., Keef, T., Cermelli, P., Salthouse, D., Twarock, R. & Zanzotto, G. (2012).Proc. R. Soc. A,468_{, 1452–1471.}

Janusz, G. & Rotman, J. (1982).Am. Math. Mon.89_{, 407–410.} Jones, H. (1990).Groups, Representations and Physics. Institute of

Physics Publishing.

Katz, A. (1989). InIntroduction to the Mathematics of Quasicrystals, edited by M. Jaric´. New York: Academic Press.

Kramer, P. (1987).Acta Cryst.A43_{, 486–489.}

Kramer, P. & Haase, R. (1989). InIntroduction to the Mathematics of Quasicrystals, edited by M. Jaric´. New York: Academic Press. Kramer, P. & Zeidler, D. (1989).Acta Cryst.A45_{, 524–533.} Levitov, L. & Rhyner, J. (1988).J. Phys. France,49_{, 1835–1849.} Moody, R. (2000). InFrom Quasicrystals to More Complex Systems,

edited by F. Axel, F. De´noyer & J. P. Gazeau. Springer-Verlag. Pitteri, M. & Zanzotto, G. (2002). Continuum Models for Phase

Transitions and Twinning in Crystals.London: CRC/Chapman and Hall.

Senechal, M. (1995). Quasicrystals and Geometry. Cambridge University Press.

Soicher, L. (2006).Oberwolfach Rep.3_{, 1809–1811. Report 30/2006.} Steurer, W. (2004).Z. Kristallogr.219_{, 391–446.}

The GAP Group (2013).GAP – Groups, Algorithms, and Program-ming, Version 4.7.2. http://www.gap-system.org.

Zappa, E., Indelicato, G., Albano, A. & Cermelli, P. (2013).Int. J. Non-Linear Mech.56_{, 71–78.}

Figure 5

Algorithm 4.

Figure 6