- 1. Introduction
- 2. Multilattices and the master equation
- 3. The master equation as a system of linear equations
- 4. Arithmetic equivalence
- 5. Applications: construction of all inequivalent multilattices with a given point group
- 6. Conclusions
- A1. Proof of Proposition 2[link]
- A2. The master equation as a linear system: §3.1[link]
- A3. Proof of Proposition 3[link]
- A4. Proof of Proposition 4[link]
- A5. Tensor form of the master equation
- A6. Proof of Proposition 5[link]
- A7. Proof of Corollary 1[link]
- References

- 1. Introduction
- 2. Multilattices and the master equation
- 3. The master equation as a system of linear equations
- 4. Arithmetic equivalence
- 5. Applications: construction of all inequivalent multilattices with a given point group
- 6. Conclusions
- A1. Proof of Proposition 2[link]
- A2. The master equation as a linear system: §3.1[link]
- A3. Proof of Proposition 3[link]
- A4. Proof of Proposition 4[link]
- A5. Tensor form of the master equation
- A6. Proof of Proposition 5[link]
- A7. Proof of Corollary 1[link]
- References

## research papers

## An algorithm for the arithmetic classification of multilattices

^{a}Department of Mathematics, York Center for Complex Systems Analysis, The University of York, Heslington, York YO10 5DD, England^{*}Correspondence e-mail: giuliana.indelicato@york.ac.uk

A procedure for the construction and the classification of monoatomic multilattices in arbitrary dimension is developed. The algorithm allows one to determine the location of the points of all monoatomic multilattices with a given symmetry, or to determine whether two assigned multilattices are arithmetically equivalent. This approach is based on ideas from integral matrix theory, in particular the reduction to the Smith normal form, and can be coded to provide a classification software package.

Keywords: monoatomic multilattices; integral matrix theory.

### 1. Introduction

A monoatomic (*N*+1)-lattice is a set of points in that is the union of *N*+1 identical Bravais lattices, and can be described by a reference (or skeletal) and *N* shift vectors , which represent the translations of the additional lattices with respect to the reference one. Equivalently, a monoatomic (*N*+1)-lattice can be described as a with *N* additional identical points per (*cf. e.g.* Pitteri & Zanzotto, 2003).

The symmetry of a multilattice is determined by those point-group operations of the skeletal lattice that leave the multilattice invariant, *i.e.* that interchange the additional points modulo lattice translations (Pitteri & Zanzotto, 1998, 2000; Fadda & Zanzotto, 2000, 2001*a*). A of an (*N*+1)-lattice can be identified to a triple , , such that

with a point-group symmetry of the skeletal lattice, an integral matrix that corresponds to the permutation action of on the points of the multilattice, and are lattice vectors. Working in components in a skeletal lattice basis, we can equivalently rewrite equation (1) as

with *M* = (*M*_{j}^{i}) a unimodular integral matrix in the lattice group of the skeletal lattice, , a matrix of integers representing a set of lattice translations, and the matrix whose columns are the components of the shift vectors.

To each triple (*M*,*A*,*T*) an (*N*+*n*) ×(*N*+*n*) matrix of the form

can be associated, and it turns out that the set of all triples that satisfy equation (2) for a given set of shift vectors is a group under matrix multiplication, which is isomorphic to the of the multilattice, and which we refer to as the lattice group of the multilattice (Pitteri & Zanzotto, 1998).

We denote by the group of all matrices of the form (3) for arbitrary unimodular integral *M*, a linear representation of a permutation *A*, and an integral matrix *T*: two (*N*+1)-lattices are arithmetically equivalent if their lattice groups are conjugated in . This notion of equivalence generalizes to multilattices the usual arithmetic classification of simple lattices in Bravais types (Schwarzenberger, 1972; Miller, 1972; Engel, 1986; Pitteri & Zanzotto, 1998).

We refer to (1) as the master equation of the multilattice. It can be used either to compute the shift vectors , given the lattice group, or to compute the lattice group given the skeletal lattice and the shift vectors.

In this work we describe a procedure to solve the master equation for any given skeletal lattice. The procedure is based on ideas from integral matrix diagonalization (Smith, 1861; Newman, 1972; Gohberg *et al.*, 1982; Havas & Majewski, 1997; Dumas *et al.*, 2001; Jäger, 2005) and automatically yields a single representative for each arithmetical equivalence class of multilattices. The idea is as follows: rewriting equation (2) as a linear system of the form

with *L* an integral matrix and a suitable unknown vector, it is a well known result that *L* can be written in a *D* whose only nonzero entries are integers, are along the diagonal and are arranged in a sequence such that each element divides the next one. Using the of *L*, the system (2) decouples into a finite number of elementary equations with integral coefficients and unknowns *X*_{i}:

whose solutions have the form

where are integers and *t*_{j} are real numbers in [0,1) [*cf.* (14)]. Hence, the transformation to Smith allows all solutions of equation (2) to be constructed at the sole cost of computing the itself.

Further, as a side result, this approach yields a simple criterion for the arithmetic equivalence of two given monoatomic multilattices whose underlying skeletal lattices are arithmetically equivalent.

In conclusion, the procedure described in this paper provides a basis for an algorithm for the classification of multilattices with an arbitrary number of points, but also yields a simple method to determine regular sets of points in arbitrary dimensions. This sort of calculation is useful for instance when high-dimensional crystallography is used, *via* a projection approach, to study quasicrystals or sets of points with (Indelicato, Cermelli *et al.*, 2012).

Also, arithmetic equivalence, which yields a finer classification than the classical classification according to affine equivalence classes of space groups, is an essential tool in characterizing and studying reconstructive phase transitions based on the notion of Bain strain, in which there is no symmetry reduction between the parent and product phases (for instance simple cubic and body- or face-centered cubic), but their lattice groups are not arithmetically equivalent (Indelicato, Cermelli *et al.*, 2012; Indelicato, Keef *et al.*, 2012).

In order to improve readability, we have collected all the proofs in Appendix *A*, and we have devoted the last section to a detailed discussion of two specific examples in three dimensions: the derivation of all inequivalent hexagonal 2-lattices (Fadda & Zanzotto, 2001*b*) and all inequivalent cubic 3-lattices (Hosoya, 1987). Such results could also be obtained using the Wyckoff positions of the relevant space groups, which can, in turn, be determined in any dimension (Fuksa & Engel, 1994; Eick & Souvignier, 2006), but our approach has the advantage of not requiring the computation of high-dimensional space groups, and taking into account arithmetical equivalence and by design.

### 2. Multilattices and the master equation

#### 2.1. Multilattices

*O*(*n*) is the orthogonal group of , is the group of integral *n* ×*n* unimodular matrices, and are the linear space and -module of *n* ×*N* real and integral matrices, respectively.

A *simple* (*Bravais*) *lattice* with basis and origin is the set of points in defined by

The *point group* of is the group of orthogonal transformations that leave the lattice invariant:

The *lattice group* of is the group of integral unimodular matrices *M* defined by (4). It follows from this definition that the lattice group is the matrix representation of the in the lattice basis.

Two lattices and are *arithmetically equivalent* if the associated lattice groups and are conjugated in , *i.e.* there exists such that

Consider now a simple lattice and *N* points not belonging to and not pairwise equivalent modulo .

An (*N*+1)-lattice with basis is the union of *N*+1 simple lattices :

The position of the points with respect to the origin of the lattice , called the skeletal lattice, is given by the shift vectors

Notice that .

The description (5) is one of many possible for a given (*N*+1)-lattice : in fact, in addition to changing the lattice basis, any relabeling of the points of the form , with a permutation of , yields an equivalent description of the same point set. The shift vectors measured with respect to the new reference lattice have the form

and are related to the original shift vectors by

#### 2.2. Essential and non-essential description of a multilattice

The lattice vectors

define a translation group that leaves the multilattice invariant, but this is not necessarily the maximal group of translational symmetries of (5). Consider a multilattice as defined in (5), with lattice vectors : we say that the description (5) is essential if all translational symmetries of belong to , *i.e.* if

When this is not the case, the set (5) is an -lattice, with , and (5) is called a non-essential description of the -lattice.

A simple criterion to establish whether a description of a multilattice is non-essential is established by Parry (2004), who proved the following result:

#### Proposition 1

Assume that the representation (5) is non-essential. Then there exist and a permutation of the set of *N*+1 integers such that

When (6) holds, decomposes into cycles of equal length, *q* say, where and .

Conversely, when (6) holds for some permutation , the representation (5) is non-essential.

This criterion implies that, for instance, a 2-lattice with shift is a simple lattice if and only if the shift is half a lattice vector of the skeletal lattice, as in the case of body-centered lattices.

#### 2.3. The lattice group of a multilattice

Loosely speaking, the symmetry of a multilattice is described by those point-group operations of the skeletal lattice that interchange the additional points modulo lattice translations. In order to make this notion precise, we need to characterize how to express permutations of the points of a multilattice in terms of the shift vectors.

The symmetric group *S*_{N+1}, acting as a group of permutations on the (*N*+1) points , also acts linearly on the -module generated by the shift vectors as follows:

We denote by the group of matrices corresponding to this action,

which is isomorphic to the symmetric group *S*_{N+1} [*cf.* pp. 309–310 of Pitteri & Zanzotto (2003), and p. 366 of Pitteri & Zanzotto (1998)]. In general, given a finite group , we refer to a group morphism as a permutation representation (permrep) of , and to the associated map as a linear permutation representation.

The symmetry of a multilattice is described by the set of triples

with a point-group symmetry of the skeletal lattice, and for , such that the action of the point-group operation on the shift vectors corresponds to a permutation of the points modulo translations of the lattice or, equivalently, to a change of descriptors of the multilattice. In short, is a

of ,Granted (4), and writing and , with , , we may rewrite equation (7) in the form

*i.e.* with *M* = (*M*_{j}^{i}), , , ,

We refer to equations (8) or (9) as the *master equation*. The matrices *M* and *A* satisfying equation (9) form the symmetry group of the multilattice.

#### Proposition 2

Given an (*N*+1)-lattice with shifts , let be the subset of of matrices *M* such that there exist and that satisfy the master equation (9). Then

We denote by the set of matrices in defined by

Proposition 2 motivates the definition of *lattice group of an* (*N*+1)-*lattice* with shift vectors *P* as the group of matrices such that

The group is isomorphic to the ).

of the multilattice, as discussed in Pitteri & Zanzotto (1998Two (*N*+1)-lattices with lattice groups and are *arithmetically equivalent* if and are conjugated in , *i.e.* if there exists a matrix such that

Further, since and are finite, they admit a finite set of generators and , with

Proposition 2 allows one to conclude that if the master equation holds for each generator, then it holds for all elements of the group . Hence equation (9), which holds for every element of , can be replaced by

#### 2.4. An example

We discuss here a two-dimensional example to show that the master equation (7) embodies the symmetries of a multilattice. Consider the monoatomic planar 3-lattice with *p*4*m**m* (Fig. 1) and square skeletal lattice: one description of this point set is obtained by letting *Q*_{0} = (0,0), *Q*_{1} = (1/2,0), *Q*_{2} = (0,1/2), and choosing the shift vectors as

A different description arises by choosing , , , with shift vectors

with the transposition of 0 and 1 that fixes 2.

The 4*m**m*, and we choose as generators of the lattice group the integral matrices

The generator *M*^{(1)} fixes *Q*_{0} and permutes *Q*_{1} and *Q*_{2} modulo the lattice, while the action of *M*^{(2)} on the points *Q*_{0}, *Q*_{1}, *Q*_{2} is lattice invariant:

and

where and are the basis vectors of the square lattice. Hence, the action of the , in terms of the matrices

of the skeletal lattice on the shifts can be written in the form (7)Alternatively, using the description of the multilattice in terms of the shift vectors , we have

and

which now involves the matrices

It turns out that these matrices are conjugated to *A*^{(1)} and *A*^{(2)} by the element of associated with the permutation : the two descriptions lead to different, but equivalent, forms of the master equation.

### 3. The master equation as a system of linear equations

The master equation is both a relation that uniquely characterizes the lattice group of a multilattice, given the shift vectors , and an equation in the unknowns , that allows all the multilattices with a given lattice group to be determined. In this section we take the latter point of view, and assume that , or rather , is given. Specifically, the problem we want to solve is:

#### 3.1. The solution procedure

The system of master equations (11), corresponding to the *K* generators of the lattice group , can be written in compact form as a linear system,

where the vectors , have components obtained by ordering lexicographically the columns of *P* and *T*^{(k)}, and *L* is an integral matrix in , whose explicit form in terms of the generators of is given in Appendix A2.

Consider first a diagonal system of linear equations with integral coefficients

with () and *D*^{J}_{i} = 0 for , and , *i.e.*

with^{1} *r* = rank *D* and *D*^{i}_{i} are integers. The set of the *m*-tuples of the form

where are integers and *t*_{j} are real numbers in [0,1), parametrizes all solutions of equation (13).

#### Proposition 3

The solutions *X* of equation (13) have the form *X* = *K*+*Y*, with and , and, conversely, all vectors of this form are solutions.

Actually, in order to find a set of representatives of the solutions in , it is enough to solve equation (13) for *S* in the set

Consider now the full system of linear equations (12): instead of solving it for a fixed value of the right-hand side, we look for solutions for some integral vector , and rewrite equation (12) in the form

Recall that *L* is a matrix with integral entries: it is a classical result that every such matrix can be reduced to a diagonal the Smith (Newman, 1972; Gohberg *et al.*, 1982). Precisely, for every matrix there exist matrices and such that

with *D*^{I}_{a} = 0 for , and *D*_{i}^{i} divides *D*_{i+1}^{i+1} if . The Smith *D* is unique, whereas the matrices *U* and *V* are not.

Notice that if is a solution of equation (15) so also is , with an arbitrary integral vector. Hence, we may restrict to solutions in [0,1)^{nN} and introduce the set

where is defined as in (14) with *m* = *n**N*, *V* is defined in (16) and, for , is the vector whose components are the integer parts of the components of *W*. In other words, is the inverse image of by *V*, translated into the of the skeletal lattice. Notice that since is a set of solutions of (13), then trivially is a set of solutions of (15). It can be proved that the definition of is independent of the choice of the diagonalizing matrices *U*,*V*.

The following results characterize completely the solution set of the master equation (15).

#### Proposition 4

Let , and *D* its Smith normal form, with *r* = rank(*D*): then all solutions of equation (15) belong to . More precisely, the system (15) admits () solutions modulo , each depending on *n**N*-*r* real parameters, and these are given by

with *V* such that *L* = *U**D**V* [with and ] and

By construction, the matrix *L* only depends on the group and its permutation representation . Every solution of the master equation (15) defines a (possibly non-essential) (*N*+1)-lattice with lattice group , as defined in equation (10), where the translation matrix is computed from .

The question arises naturally as to whether two solutions of the same master equation correspond to arithmetically equivalent multilattices. We shall discuss this topic in the following section.

### 4. Arithmetic equivalence

The main result in this section shows that two equivalent multilattices have the same Smith normal form, and provides a criterion to establish when two multilattices are equivalent.

Consider two equivalent (*N*+1)-lattices. By definition, their lattice groups and are conjugated by some

In particular, the associated subgroups and of the lattice group of the skeletal lattice, as well as their permutation representations in , are conjugated by *H* and *B*, respectively. To simplify, we choose the generators

of and to be pairwise conjugate, which implies in turn that

for every .

We write the master equations corresponding to each multilattice as in equation (12),

with Smith canonical form

Finally, for a given square matrix , we denote by the square matrix of the form

#### Proposition 5

For two equivalent (*N*+1)-lattices, *L* and satisfy the relation

where is the integral matrix associated with the conjugating matrices *H* and *B* in (19) through the relation (36) in Appendix A. As a consequence, the matrices *L* and in equation (20) have the same Smith normal form

Further, the vectors in equation (21) are related through

with are such that *L* = *U**D**V* and , and is a vector of integers.

Conversely, given two non-necessarily equivalent (*N*+1)-lattices, assume that the groups and defined in Proposition 2, as well as their permutation representations in , are conjugated, *i.e.* there exists and such that equation (19) holds for some set of generators, and therefore (22) holds. If there exists an integral vector such that (23) holds, then the two multilattices are equivalent.

Notice that, as we will see below, there exist multilattices for which (23) is not true, that have the same associated Smith normal form but are not equivalent.

The above result allows one, among other things, to classify the inequivalent solutions of the master equation, as shown by the following corollary. Consider to this purpose a group with generators , and a permutation representation , and write for the images of the generators of . Let *L* be the integral *n**N**K* ×*n**N* matrix associated with these generators, and let *D* = *U*^{-1} *L* *V*^{-1} be its Smith normal form.

#### Corollary 1

Under the above hypotheses, consider two solutions *X* and of the master equation in diagonal form , and let *S* = *D**X*, . Then the corresponding multilattices are arithmetically equivalent if and only if there exists an integral vector such that

where is the integral matrix associated with the conjugating matrices *H* and *B* through the relation (36), with elements of the centralizers of and its permutation representation, respectively, *i.e.*

for every .

The above criterion for arithmetic equivalence could also be formulated in terms of the integral matrices *T* and , but we find it easier to use it in this form, as the subsequent examples show.

### 5. Applications: construction of all inequivalent multilattices with a given point group

The procedure discussed in the previous sections can help to solve a classical problem of the arithmetic classification of multilattices, namely how to generate all arithmetic equivalence classes of (*N*+1)-lattices with a given Notice that the algorithm in §3 involves the lattice group of the skeletal lattice, instead of its this is necessarily so since two skeletal lattices with the same could be arithmetically inequivalent, and have therefore lattice groups that are not conjugated in , as is the case for the three cubic lattices in three-dimensions (primitive, face centered and body centered).

#### 5.1. First example: 2-lattices with hexagonal in three dimensions

We show how to obtain all inequivalent 2-lattices with hexagonal 6/*m**m**m* and space groups *P*6_{3}/*m**m**c* and *P*6/*m**m**m* (Nos. 194 and 191 in *International Tables for Crystallography* Volume *A*). These structures are listed as 6, 27 and 28 in Fadda & Zanzotto (2001*b*).

In this case *n* = 3, *N* = 1 and *K* = 3. The hexagonal has the : there is only a single arithmetic class in this case, and the corresponding lattice group is the matrix representation of the in the lattice basis. Using the conventional choices for the lattice basis given in *International Tables for Crystallography* Volume *A* (Hahn, 2005), we choose as generators of the integral matrices (Fadda & Zanzotto, 2001*b*)

together with the inversion, denoted here as *M*^{(3)}. In this case, all possible representations of 6/*m**m**m* as a permutation group on 2 elements result by associating to each generator *M*^{(i)} either the identity permutation or the transposition, corresponding to *A*^{(i)} = 1 or *A*^{(i)} = -1, respectively.

We describe below only the two cases that yield non-trivial results.

#### 5.2. Second example: 3-lattices with cubic in three dimensions

We discuss here an application to 3-lattices, showing how to obtain the structures with three identical atoms per ), p. 16, corresponding to the space groups , and (Nos. 221, 225 and 229, respectively, in *International Tables for Crystallography* Volume *A*). According to the classification of Hosoya, such structures belong to genus *A*_{3} (three identical atoms per unit cell).

The work can be organized following the steps listed in §3, with *n* = 3 and *N* = 2: fix one of the three cubic lattices in , consider its lattice group, which is conjugate to the cubic *O*_{h}, determine all its permutation representations, write the master equation and solve it with the techniques described in the paper.

As a first step we compute all permutation representations of *O*_{h}, recalling that they can be determined in terms of its actions on the spaces *O*_{h}/*H*, with *H* a maximal (Aschbacher, 2000).

Since we are interested in permutation representations on sets of three objects, we only need to consider subgroups of *O*_{h} of index less or equal to three, namely *D*_{4h} (index 3), *T*_{d} (index 2), *T*_{h} (index 2) and *O* (index 2).

We use here a presentation of *O*_{h} in terms of five generators (*K* = 5):

The permutation representations corresponding to the maximal subgroups of *O*_{h} are

### 6. Conclusions

Monoatomic multilattices are periodic structures that generalize simple lattices in any dimension. Their study is important not only for materials science, but also to provide a general description of those quasiperiodic structures that can be obtained by projection of regular sets of points from high- to low-dimensional spaces, *via*, for instance, the well known cut-and-project scheme for quasicrystals.

A first fundamental problem is to establish whether two multilattices are equivalent in some sense, as well as to determine all multilattices that belong to a given equivalence class. In this context, it has been proved that, in analogy to simple lattices, arithmetic equivalence is strictly finer than affine equivalence (Pitteri & Zanzotto, 1998). Hence, we focus here on arithmetic equivalence.

We approach the problem *via* the so-called master equation (1), that either characterizes all monoatomic multilattices with a given symmetry or can be used to establish the symmetry group of a given multilattice. By reducing the master equation to a suitable normal form,* i.e.* the Smith normal form, it is possible to enumerate all solutions, and determine easily which of these solutions are arithmetically equivalent using the criterion in Proposition 1, which only involves the characterization of the of a finite crystallographic group. Since the centralizers of the crystallographic groups in any dimension are finite or finitely generated, this procedure yields an algorithm which, in principle, can be coded and yields a solution to the arithmetic classification problem for multilattices.

In order to elucidate the basic features of our method, we discuss two examples from the literature, recovering in a few steps some relevant cubic and hexagonal 2- and 3-lattices in three dimensions.

### APPENDIX A

### Proofs

#### A1. Proof of Proposition 2

By hypothesis, if , there exist and *T*_{M},*T*_{H} integral matrices such that

Hence

Further, by multiplying *M**P* = *P**A*_{M}+*T*_{M} to the left by *M*^{-1} and to the right by *A*_{M}^{-1}, we find

Hence, since *T*_{M}*A*_{H} +*M**T*_{H} and *M*^{-1}*T*_{M}*A*_{M}^{-1} are matrices of integers, *M**H* and *M*^{-1 } satisfy the master equation, and is a group. Further, the mapping is single-valued. In fact, noting first that it is implicit in the hypothesis that the shift vectors *P* provide an essential description of the multilattice, assume that there exists such that *P* = *P**A*+*T*, *i.e.* , where *I*_{n} and *I*_{N} are the identity in and , respectively. Explicitly, this means that

with the permutation corresponding to *A*, and this, by Proposition 1, implies that the description is non-essential, which is a contradiction. Hence, the map is a group morphism, and *A*_{MH} = *A*_{M}*A*_{H}. Finally, the above argument shows that the map

is also a group morphism, so that is also a group.

#### A2. The master equation as a linear system: §3.1

The master equation (8) for a fixed element

can be rewritten as a conventional system of linear equations. To do so, given and , define

so that *a* takes values in . Conversely, let and define and *i* through the identities

where [·] denotes the integer part of its argument. As *a* varies in , then and *i* take values in and , respectively, and the relation between *a* and the pair is bijective. Let

*i.e.*

with *I*_{n} the identity matrix in , and

where , *i* are defined as in equation (25) and, for

with and Kronecker deltas. The *n**N*-dimensional vector has components that are obtained by ordering the vectors .

In terms of the vectors and and the matrix *L*, the master equation (8) takes the form

The above assertion follows from a simple argument: let and , with , and consistent with the indexing rule (27). Then

Hence

Consider now the system of master equations (11) for the full set of generators of , *i.e.*

with *K* the number of generators of . The associated system of linear equations (28) is now replaced by a system of the form

with

with given by

with inverse

#### A3. Proof of Proposition 3

Given , then for all there exist and such that

Then *D*^{i}_{i}*X*^{i} = *D*^{i}_{i}*K*_{i} + *C*^{i} and, as a consequence, *X*^{i} = *K*^{i} + *Y*^{i} with *Y*^{i} = *C*^{i}/*D*^{i}, for , and the statement is proved.

#### A4. Proof of Proposition 4

The general procedure to solve equation (15) is as follows: let

so that, since , the system (15) can be written in the form

* i.e.*

where *r* = rank (*D*^{J}_{a}). By Proposition 3, it is sufficient to solve equation (33) in the set : we obtain

with *t*_{i} real parameters.

Once the *X*^{a} and the corresponding are computed, the right-hand sides of the master equation (29) are determined, and (30)_{2,3} yield the solution in terms of the and .

#### A5. Tensor form of the master equation

The relation between the master equation and the matrix *L* can be rewritten in more compact form as follows. For and , consider the fourth-order tensor

with components , and where *A*^{T} is the transpose of the matrix *A*. The set of tensors of the form (34) is a group with the product

and the indexing rules (24) and (25) define a morphism between the group of such tensors and the group of invertible *n**N* ×*n**N* matrices.

#### Proposition 6

For , , let

then the rule

defines a map between , with product *, and which is a group morphism.

#### Proof

Notice first that if *M* and *A* are invertible, then *W* is invertible, with inverse *W*^{-1} associated with the tensor , with *A*^{-T} = (*A*^{-1})^{T}. Now let : then

which proves the assertion.

□

Further, the tensors of the form (34) act linearly on the space of real matrices as follows:

Letting be given by equation (26), the above action corresponds to the linear action of on . In fact

The tensor form of the master equation (9) then follows in the form

with *I*_{N} and *I*_{n} the *N*-dimensional and *n*-dimensional identity matrices, respectively.

#### A6. Proof of Proposition 5

Consider two mutually conjugated generators of and ,

by hypothesis

with *Q* given by (18), so that, in particular, and . Letting

then

The first assertion of the thesis then follows by letting *W* be the matrix in associated with through the rule (36), and using equation (39) and the definition (30) of *L*.

Further, for each *k*, equations (37) and (38) imply that

which in turn means that

with the integral vector associated with the matrix *H*^{-1}*R* through the relation (30)_{2} [notice that and ]. Finally, we obtain equation (23) by multiplying the above identity by .

#### A7. Proof of Corollary 1

We first need an auxiliary result.

#### Proposition 7

Given two (*N*+1)-lattices as above, assume that and , subgroups of the lattice group of the skeletal lattices, as well as their permutation representations in , are conjugated, *i.e.* there exist and such that writing

for the generators of and , respectively, then

for each . If there exists an integral vector such that

with the same notations of Proposition 5, the two multilattices are equivalent.

#### Proof

Clearly, (23) implies that there exists such that

with the integral vector associated with the matrix *H*^{-1}*R* through the relation (30)_{2}. This, together with equation (19), implies in turn that

holds for each *k*, with *Q* given by equation (18).

□

In order to prove Corollary 1, it is enough to apply Proposition 7, with and . In this case, the conjugants *H* and *B* are just operations that fix and , respectively, *i.e.*, elements of the centralizers.

### Footnotes

^{1} must be zero in order that equation (13) has solutions.

### Acknowledgements

The author acknowledges valuable discussions with P. Cermelli and G. Zanzotto. This work was supported by The Leverhulme Trust Research Leadership Award F/00224 AE, the Marie Curie IEF-FP7 project MATVIR, the MATHMAT Project of the University of Padova and the PRIN project 2009 `Mathematics and Mechanics of Biological Assemblies and Soft Tissues'.

### References

Aschbacher, M. (2000). *Finite Group Theory*, 2nd ed. Cambridge University Press.

Dumas, J., Saunders, B. & Villard, G. (2001). *J. Symb. Comp.* **32**, 71–99. Web of Science CrossRef

Eick, B. & Souvignier, B. (2006). *Int. J. Quantum Chem.* **106**, 316–343. Web of Science CrossRef CAS

Engel, P. (1986). *Geometric Crystallography: An Axiomatic Introduction to Crystallography*. Dordrecht: Kluwer Academic Publishers.

Fadda, G. & Zanzotto, G. (2000). *Acta Cryst.* A**56**, 36–48. Web of Science CrossRef CAS IUCr Journals

Fadda, G. & Zanzotto, G. (2001*a*). *Acta Cryst.* A**57**, 492–506. Web of Science CrossRef CAS IUCr Journals

Fadda, G. & Zanzotto, G. (2001*b*). *Int. J. Non-Linear Mech.* **36**, 527–547. Web of Science CrossRef

Fuksa, J. & Engel, P. (1994). *Acta Cryst.* A**50**, 778–792. CrossRef CAS Web of Science IUCr Journals

Gohberg, I., Lancaster, P. & Rodman, L. (1982). *Matrix Polynomials*. New York: Academic Press.

Hahn, Th. (2005). *International Tables for Crystallography*, Vol. A, 5th ed. Heidelberg: Springer.

Havas, G. & Majewski, B. S. (1997). *J. Symb. Comp.* **24**, 399–408. CrossRef Web of Science

Hosoya, M. (1987). *Bull. College Sci. Univ. Ryukyus*, **44**, 11–74. CAS

Indelicato, G., Cermelli, P., Salthouse, D. G., Racca, S., Zanzotto, G. & Twarock, R. (2012). *J. Math. Biol.* **64**, 745–773. Web of Science CrossRef PubMed

Indelicato, G., Keef, T., Cermelli, P., Salthouse, D., Twarock, R. & Zanzotto, G. (2012). *Proc. R. Soc. London Ser. A*, **468**, 1452–1471. CrossRef

Jäger, G. (2005). *Computing*, **74**, 377–388.

Miller, W. (1972). *Symmetry Groups and their Applications*. New York: Academic Press.

Newman, M. (1972). *Integral Matrices*. New York: Academic Press.

Parry, G. P. (2004). *Math. Mech. Solids*, **9**, 411–418. Web of Science CrossRef

Pitteri, M. & Zanzotto, G. (1998). *Acta Cryst.* A**54**, 359–373. Web of Science CrossRef CAS IUCr Journals

Pitteri, M. & Zanzotto, G. (2000). *Symmetry of Crystalline Structures; a New Look at it, Motivated by the Study of Phase Transformations in Crystals*. Proceedings of the International Congress SACAM 2000, edited by S. Adali, E. V. Morozov and V. E. Verijenko, Durban, South Africa.

Pitteri, M. & Zanzotto, G. (2003). *Continuum Models for Phase Transitions and Twinning in Crystals*. Boca Raton: Chapman and Hall.

Schwarzenberger, R. L. E. (1972). *Math. Proc. Camb. Philos. Soc.* **72**, 325–349. CrossRef

Smith, H. J. S. (1861). *Philos. Trans. R. Soc. London*, **151**, 293–326. CrossRef

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