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On discrete groups of Euclidean isometries: representation theory, harmonic analysis and splitting properties

We study structural properties and the harmonic analysis of discrete subgroups of the Euclidean group. In particular, we 1. obtain an efficient description of their dual space, 2. develop Fourier analysis methods for periodic mappings on them, and 3. prove a Schur-Zassenhaus type splitting result.


Introduction
The main objective of the present contribution is a study of various aspects of discrete subgroups of the Euclidean group E(d) concerning their representations, harmonic analysis and splitting properties. Special cases of such groups are finite subgroups of the rotation group O(d) and the crystallographic space groups. While these objects are very well studied, cf. e. g. [12,6,8] for classic references, the theory for general discrete groups of Euclidean isometries is considerably less developed. In a sense that is made precise below, they interpolate and combine aspects of finite rotation groups and space groups as they might be assumed to embed into a subgroup of O(d 1 ) ⊕ S for some d 2 -dimensional space group S, where d 1 + d 2 = d. We will see that it is possible to identify a "translational part" of such a group featuring some properties analogous to those of a d 2 -dimensional lattice. This will allow us to 1. obtain an efficient description of the dual space of discrete subgroups of E(d), 2. to develop Fourier analysis methods for periodic mappings on them, and 3. to prove a Schur-Zassenhaus type splitting result with respect to the translational part.
Besides its intrinsic mathematical interest, our motivation originates in a physical question on the stability of so-called Objective Structures. These particle systems were introduced by James in [14] as a far reaching generalization of lattice systems and have been deployed successfully to describe a remarkable number of important structures ranging from biology (parts of viruses) to nanoscience (carbon nanotubes).
In order to motivate and illustrate our investigations let us consider Z d as the most basic and classical example of a lattice system. Localized mappings u ∈ ℓ 1 (Z d ) (say) can be conveniently analyzed in terms of their Fourier transform which in particular induces a resolution into plane waves e 2πi k,· with dominant contributions from small "wave numbers" |k|. So as to account for lattice mappings with significant contributions from large large wave numbers, a feasible strategy is to investigate N -periodic functions u : Z d → C, N ∈ N, for which u(x + N e i ) = u(x) for all i ∈ {1, . . . , d} in the asymptotic regime N → ∞. Such a function might alternatively be looked at as a function on Z d N and is described by its discrete Fourier transform Here the "wave vectors" k are eventually dense in the unit cell [0, 1] d as the period N becomes larger and larger. In this way one obtains a Fourier descriptions for all periodic functions which, upon truncating the maximal periodicity at finite N , lends itself to a directly controllable approximation in numerical simulations. By a simple coordinate change such an analysis applies to general lattice systems AZ d , A ∈ GL(d, R), which are easily seen to be the orbit of a single point under the action of a discrete group of translations on R d . These point sets are of paramount interest in solid state physics where they might describe positions of atoms and molecules. More generally, an Objective Structure can be defined as the orbit of a single point under the action of a general discrete subgroup of the Euclidean group E(d) on R d , cf. [14]. Those structures are thus characterized by the fact that any two points "see" an identical environment of other points, modulo a rigid motion.
One of the aims of the present contribution is to develop an extension of the above described Fourier analysis for discrete translation groups to general discrete groups of Euclidean isometries. While in principle a Fourier transform is defined on the dual space of such a group, an efficient description of these spaces appears to be missing. Moreover, the incorporation of periodic mappings with significant "long wave-length" contributions is non-trivial. In fact, due to a possible lack of periodicity, even the definition of a quantity that can be interpreted as a wave-length is not obvious. Our main goal is therefore, by exploiting the special structure of discrete subgroups of E(d), to provide an efficient and extensive description of their dual spaces. In particular we identify a finite union of convex "wave vector domains" for each such space, which reflects the existence of an underlaying translational part of finite index.
We proceed to give a more detailed account of our results and a plan of the paper. In Section 2 we first collect basic definitions and properties of the Euclidean group and space groups and cite a characterization of discrete subgroups of the Euclidean group: Up to conjugation in E(d) a discrete subgroup G of E(d) embeds into a subgroup of O(d 1 ) ⊕ S for a spacegroup S ⊂ E(d 2 ), where d 1 + d 2 = d, with surjective projection π onto S, cf. Fig. 1.
.     Whereas the spacegroup S has a rich translation group T S , this is in general not the case for G. To overcome this problem, in Section 3 we fix a section T ⊂ G of T S , which, however, will not be a group in general. We also set F = ker(π) ∩ G and analyze T and F in some detail. A first remarkable fact then is that T N (set of N -fold products of elements in T ) for suitable N ∈ N is a subgroup and even a normal divisor, see Theorem 3.11, which is essential in our later considerations on periodic functions.
In Section 4 we obtain our main result on the structure of representations of G. We consider the subgroup T F of G, which corresponds to the set of preimages of T S under π, and analyze in detail the representations of T F and the induced representations of T F in G. In Theorem 4.16 we prove that the latter decompose into a finite disjoint union , for a suitable finite R ⊂ T F. This in particular allows for an interpretation of ρ∈R R d 2 /G ρ as a set of "wave vectors". A version for periodic representations is given in Theorem 4.17 where the orbit spaces R d 2 /G ρ are replaced by suitable rescalings of L * S , the dual lattice of translations in S. As a result of Theorem 4.16, we obtain that -up to a negligible setthe whole dual space G is equal to the same set of induced representations. More precisely, for each ρ ∈ R there is a zero-set N ρ ⊂ R d 2 /G ρ (with respect to the Lebesgue-measure) and a zero-set N ⊂ G (with respect to the Plancherel-measure on G) such that the above mapping is bijective when restricted to cf. Theorem 4.19. We remark that, as compared to general results in this direction obtained with the Mackey machine (see in particular [19,16] and also cp. Theorem A.10 below), we obtain an explicit labeling of representations with "wave vectors" in a finite union of convex domains in R d 2 , which in turn are identified as fundamental domains of associated space groups.
In the following, comparatively elementary Section 5 we introduce an inner product space of functions that satisfy a suitable periodicity assumption. We then proceed to develop a harmonic analysis on such objects by defining the Fourier transform for both periodic and absolutely summable functions and formulating well-known theorems like the Plancherel formula within our setting.
In Section 6 we then address the question if G splits into a translational part (more precisely, some T m ⊳ G) and a finite complement. In general this is not the case. However, our main structural splitting result 6.7 provides such representations as semidirect products for quotient groups of G with respect to a series of eventually sparse normal subgroups: If in addition n ∈ N is coprime to m and |S/T S |, then there is a group H ≤ G/T nm such that (Note that for space groups and m = 1 such a result is mentioned in [3,10].) Finally, Section A in the appendix collects some well-known definitions and theorems of harmonic analysis like the definition and basic properties of dual spaces and induced representations.
We close this introduction with an outlook to applications on the stability analysis of particle systems. Assuming that particles at different sites interact, one is naturally led to the question if an Objective Structure corresponds to a stable configuration. Similar questions are by now well understood in lattice systems, see, e. g., [13]: At equilibrium configurations the second order Taylor approximation of the configurational energy is conveniently analyzed in Fourier space and formulae for stability constants under rather generic interaction assumptions are available [7]. The results in the current contribution will indeed allow for an analogous characterization of stability constants for Objective Structures. This will be realized in the forthcoming contributions [24,25], where we provide a characterization that even leads to a numerical algorithm for testing the stability of a given structure and to novel applications to nanotubes.

Notation
We will use the following notation. For all groups G and subsets S 1 , S 2 ⊂ G we denote S 1 S 2 := {s 1 s 2 | s 1 ∈ S 1 , s 2 ∈ S 2 } ⊂ G the product of group subsets. For all S ⊂ G, n ∈ Z and g ∈ G we denote S n := {s n | s ∈ S} ⊂ G and gS := {gs | s ∈ S} ⊂ G.
For two groups G, H we write H < G if H is a proper subgroup of G and H ⊳ G if H is a normal subgroup of G. For a subset S of a group G we write S for the subgroup generated by S.
Moreover, let N be the set of all positive integers {1, 2, . . . }, Z n be the group Z/(nZ), e i be the i th standard coordinate vector (0, . . . , 0, 1, . . . , 0) ∈ R d and I n ∈ R n×n be the identity matrix of size n. We use capital letters for matrices. For A = (a ij ) ∈ C m×n and B = (b ij ) ∈ C p×q , their direct sum and their Kronecker product are respectively. The Hermitian adjoint of A is denoted A H .

Discrete groups of Euclidean isometries
This preliminary section serves to collect some basics on the Euclidean group acting on R d and of its discrete subgroups. We also introduce some general notation.

The Euclidean group
Let d ∈ N be the dimension. We denote the set of all Euclidean distance preserving transformations of R d into itself by the Euclidean group E(d). The elements of E(d) are called Euclidean isometries. It is well-known that the Euclidean group E(d) can be described concretely as the outer semidirect product of R d and O(d), the orthogonal group in dimension d: The group operation is given by Moreover, we define the maps We call a set of translations linearly independent if their translation components are linearly independent. The natural group action of E(d) on R d is given by In this contribution we use a calligraphic font for subsets and particularly for subgroups of E(d). For every group G < E(d) we denote the orbit of a point x ∈ R d under the action of the group G by We is decomposable, i. e., there is a decomposition of R n+1 into the direct sum of two proper If this is not the case, the discrete group G is called indecomposable, see, e. g., [8,Appendix A.3]. An indecomposable discrete group G < E(d) is also called a (d-dimensional) space group. Below we also present a (wellknown) characterization of the space groups and the decomposable discrete subgroups of E(d), respectively, which does not use representations.
In the physically important case d = 3, all space groups and discrete decomposable subgroups of E(3) are well-known and classified, see, e. g., [2] and [22], respectively.

Space groups
The following theorem is well-known, see, e. g., [8,Appendix A.3]. Also the following theorem is well-known. Proof. This is clear by Theorem 2.2(i).

Discrete subgroups of the Euclidean group
We recall that two subgroups G 1 , G 2 < E(d) are termed conjugate subgroups under the group E(d) if there exists some g ∈ E(d) such that g −1 G 1 g = G 2 . Note that every such conjugation corresponds to a coordinate transformation in R d . Now we may state the following well-known characterization of the discrete subgroups of E(d). For this purpose for all d 1 , d 2 ∈ N we define the group homomorphism Proof. Let d ∈ N be the dimension and G < E(d) be discrete. If G is a space group, the assertion is trivial. If G

Translational sections
Our first aim will be to efficiently describe the discrete group G < E(d) in terms of the range S and the kernel F of π| G . An important step will be to fix and analyze a section T ⊂ G of the translation group T S of S. The main result in this section is Theorem 3.11 which characterizes m ∈ N for which T m ⊳ G.
(iii) The quantities d, d 1 , d 2 , F, S and T S are uniquely defined by G. In general for given G there is no canonical choice for T , see Example 3.4.
(iv) Let G be given. In general, for every choice of T the set T is not a subset of Trans(d), see Example 3.3. Moreover, in general for every choice of T the set T is not a group and the elements of T do not commute, see Example 3.5.
(v) Let G be given. One possible choice for T is the following. Let For the following examples for all angles α ∈ R we define the rotation matrix

Example 3.3 (Helical groups)
. Let d 1 = 2, d 2 = 1, α ∈ R be an angle, n ∈ N, Then T is isomorphic to Z, F is a cyclic group of order n, P is a group of order 2 and FP a dihedral group of order 2n. Moreover, T , T F, T P and T FP are decomposable discrete subgroups of E(3). If we have α ∈ R \ (2πQ), the groups T , T F, T P and T FP are so-called helical groups, i. e. infinite discrete subgroups of the Euclidean group E(3) which do not contain any translation except the identity.
Then the choice R(π/2) ⊕ t ∈ T as well as R(3π/2) ⊕ t ∈ T is possible. In particular, the choice of T is not unique.

Example 3.5.
We present a discrete group G < E(8) such that for every choice of T the set T is not a group and the elements of T do not commute.
and π : G → S be the natural surjective homomorphism with kernel Thus, the elements of T do not commute. Now we suppose that T is a group. Since π −1 (id E(2) ) = F and by (1), we have π −1 (id E(2) ) ⊂ T . This contradicts the claim that π| T is bijective. Thus, T is not a group.
For the remainder of this section we fix the dimension d ∈ N, the discrete group G < E(d) and the quantities d 1 , d 2 , T , F, S, T S as introduced by Definition 3.1. The following lemma collects some elementary properties.

Lemma 3.6. (i) The group F is finite.
(ii) For all n ∈ N the set T n F is independent of the choice of T , and it holds In particular, it holds T F ⊳ G.
is the natural surjective homomorphism with kernel F. In particular, G/T F is finite.
In particular, the group T F/F is commutative.
(v) For all n ∈ Z \ {0} the map T → T n , t → t n is bijective.
Proof. Let π : G → S be the natural surjective homomorphism with kernel F.
Thus, the group F is finite.
(ii) Let n ∈ N. The set T n F is the preimage of T n S under π. Since T n S is a normal subgroup of S, the set T n F is a normal subgroup of G.
(iii) This is clear, since T F is the preimage of T S under π.
(iv) Let n ∈ N. Since T S is isomorphic to Z d 2 , the map ϕ 1 : T S → T n S , t → t n is a group isomorphism. Since F is the kernel of π and T n F the preimage of T n S under π, the map ϕ 2 : T n F/F → T n S , gF → π(g) is an isomorphism. This implies the assertion, i. e. the map ϕ −1 2 • ϕ 1 is an isomorphism.
(v) Let n ∈ Z \ {0}. The map ψ : T → T n , t → t n is surjective. Since the map T S → T n S , t → t n is injective, the map ψ is injective and thus, bijective.
If T is a group, it is naturally isomorphic to T S .
Furthermore, for all n ∈ Z it holds Proof. Let m ∈ Z\{0} such that T m is a group. Let π : T F → T S be the natural surjective homomorphism with kernel F. Let ϕ be the inverse function of π| T , i. e. ϕ : T S → T is the canonical bijection. The map is an isomorphism. Since T m is a group, the map We proceed to show that, albeit T is not a group in general, the situation is much better for special powers of T . Proof. Let m ∈ M 0 , t ∈ T and g ∈ T F. By Lemma 3.6(iv) there exists some f ∈ F such that gt m = t m gf. Proof. Since F is a normal subgroup of G, for all g ∈ G the map is a group automorphism. Let n be the order of the automorphism group of F. For all g ∈ G it holds ϕ n g = id. Thus for all g ∈ G and f ∈ F we have i. e. g n and f commute. Now we show that for all g, h ∈ T F the elements g n|F| and h commute.
Now we show that T n|F| 2 is a subgroup of T F. Let t, s ∈ T . We have to show that t n|F| 2 s −n|F| 2 ∈ T n|F| 2 . Let r ∈ T and f ∈ F such that ts −1 = rf . Since T F/F is commutative, there exists some e ∈ F such that t n|F| s −n|F| = r n|F| e. By (3) and (2) we have Since T n F is a normal subgroup of G, there exist some s ∈ T and f ∈ F such that g −1 t n g = s n f.
The following Theorem is a key observation on the structural decomposition of G. Not only do we have the existence of exponents m ∈ N such that T m ⊳ G as guaranteed by Lemma 3.10, but the set of such 'good' exponents will in fact be of the form m 0 N, m 0 ∈ N. Proof. We define the set First we show that M 0 is a subgroup of the additive group of integers Z. It is clear that 0 ∈ M 0 . Let n 1 , n 2 ∈ M 0 . We have to show that n 1 − n 2 ∈ M 0 . Let ϕ : T S → T be the canonical bijection. By Proposition 3.9 and Lemma 3.7, for all t, s ∈ T S it holds and thus, T n 1 −n 2 is a group. It remains to show that T n 1 −n 2 is a normal subgroup of G. Without loss of generality we assume that n 1 , n 2 = 0, i. e. n 1 n 2 = 0. Let g ∈ G and t ∈ T . Since T n 1 , T n 2 ⊳ G, there exist some s 1 , s 2 ∈ T such that gt n 1 g −1 = s n 1 1 and gt n 2 g −1 = s n 2 2 . Since s n 1 n 2 1 = gt n 1 n 2 g −1 = s n 1 n 2 2 and the map T → T n 1 n 2 , r → r n 1 n 2 is bijective, it holds s 1 = s 2 . Now we have By Lemma 3.10 and since M 0 ⊂ M 0 , the group M 0 is nontrivial. Since every nontrivial subgroup of Z is equal to nZ for some n ∈ N, there exists a unique m 0 ∈ N such that

Wave vector characterization of the dual space
We now study representations of T F and their induced representations on G. As T F = π −1 (T S ), where π : G → S is the natural surjective homomorphism, we are led to first considering characters on T F by lifting those on T S via π −1 . (The reader is referred to Section A in the appendix for basic material on representations of a discrete subgroup H of E(d), its dual space H and representations induced by a normal subgroup of finite index.) where π : T F → T S is the natural surjective homomorphism.
Since T F is a normal subgroup of G, G acts on T F , cf. Definition A.1.

Lemma 4.2.
For all g ∈ G and k, k ′ ∈ R d 2 it holds where π : G → S is the natural surjective homomorphism.
Proof. Let g ∈ G, k, k ′ ∈ R d 2 and π : G → S be the natural surjective homomorphism. For all h ∈ T F it holds In our analysis of periodic representations below it will necessary to analyze representations that are trivial on T n , n ∈ N. To this end, we recall that a set L ⊂ R n is a lattice if L is a subgroup of the additive group R n which is isomorphic to the additive group Z n , and which spans the real vector space R n . The dual lattice L * (also called the reciprocal lattice) of a lattice L ⊂ R n is the set This is indeed a lattice as well, see, e. g., [20,Chapter 1].
In view of G and the characters of T F acting on T F we proceed to introduce the following equivalence relation.

Definition 4.5.
We define the relation ∼ on T F by Remark 4.6. One can also define an equivalence relation ∼ on the set of all representations of T F by for all representations ρ, ρ ′ on T F. Proof. It is clear that ∼ is reflexive. Now we show that ∼ is symmetric. Let ρ, ρ ′ ∈ T F such that ρ ∼ ρ ′ . There exist some g ∈ G and k ∈ R d 2 such that g · ρ = χ k ρ ′ . This implies where π : G → S is the natural surjective homomorphism. Now we show that ∼ is transitive. Let ρ, ρ ′ , ρ ′′ ∈ T F such that ρ ∼ ρ ′ and ρ ′ ∼ ρ ′′ . There exist some g, g ′ ∈ G and k, where π : G → S is the natural surjective homomorphism.
where π : T F → (T F) m is the natural surjective homomorphism, is bijective. In particular, the set {ρ ∈ T F | ρ| T m = I dρ } is finite.
(iii) Let K be a representation set of (L * S /m)/L * S and P be a representation set of G/(T F). Then, for all ρ, ρ ′ ∈ {ρ ∈ T F |ρ| T m = I dρ } it holds Proof. Let m ∈ N such that M 0 = mN.
(i) Let R be a representation set of {ρ ∈ T F | ρ| T m = I dρ }/∼. We have to show that for all ρ ∈ T F there exists some ρ ′ ∈ R such that ρ ∼ ρ ′ . Let ρ ∈ T F. By Proposition 3.9 the group T m is a subgroup of the center of T F and thus, by Proposition A.3 for all t ∈ T m there exists some λ ∈ C such that |λ| = 1 and ρ(t) = λI dρ . Hence, there exists some one-dimensional representation χ ∈ T m such that ρ| T m = χI dρ .
There exists some k ∈ R d 2 such that χ| T m = χ k | T m : By Lemma 3.7 the group T m is isomorphic to Z d 2 . Thus, there exist t 1 , . . . , t d 2 ∈ T m such that {t 1 , . . . , t d 2 } generates T m . For all j ∈ {1, . . . , d 2 } there exists some α j ∈ R such that exp(2πiα j ) = χ(t j ). For (ii) This is clear by Proposition A.4 and Remark 3.12(ii).
where we used (14) in the first step and that χ k−l = 1 since k − l ∈ L * S in the last step. The other direction of the assertion is trivial. Proof. This is clear by Lemma 4.9.
Definition 4.11. For all ρ ∈ T F we define the set where π : G → S is the natural surjective homomorphism.

Proposition 4.12. For all ρ ∈ T F the set G ρ is a space group and it holds
Proof. Let ρ ∈ T F and m ∈ N such that M 0 = mN. First we show that G ρ is a subgroup of E(d 2 ). Let g 1 , g 2 ∈ G ρ . We have to show that be the group of all translations of G ρ . It is clear that There exists some g ∈ G such that g · ρ = χ k ρ and L(π(g)) = I d 2 . The latter implies π(g) ∈ T S and thus, g ∈ T F. By (14) we have ρ = χ k ρ. Letρ be a representative of ρ. There exists some T ∈ U(d ρ ) such that T Hρ (g)T = χ k (g)ρ(g) for all g ∈ T F. Moreover, by Proposition 3.9 the set T m is a subset of the center of T F and hence, by Proposition A.3ρ(g) is a scalar multiple of I dρ for all g ∈ T m . Hence, we have χ k (g) = 1 for all g ∈ T m and k ∈ L * S /m by Lemma 4.4. Now we show that L * S ≤ τ (H). Let k ∈ L * S . By Lemma 4.4 we have χ k | T = 1. Since we also have χ k | F = 1, we have χ k = 1. Thus we have id G · ρ = χ k ρ and (I d 2 , k) ∈ H, i. e. k ∈ τ (H). Now we show that G ρ is discrete. Since τ (H) is a subgroup of L * S /m, the group H is discrete. Since L(G ρ ) is a subgroup of the finite group L(S), the index |G ρ : H| = |L(G ρ )| is finite. Hence, the group G ρ is discrete (see, e. g., [22,Theorem 7.1]). Since L * S is a subgroup of τ (H), the group G ρ contains d 2 linearly independent translations. By Theorem 2.1 the group G ρ is a space group.

Lemma 4.13. For all N ∈ M 0 and ρ ∈ T F such that ρ|
Proof. Let N ∈ M 0 and ρ ∈ T F such that ρ| T N = I dρ . Let k ∈ L * S /N and g ∈ G ρ . We have to show that g · k ∈ L * S /N . Let π : G → S be the natural surjective homomorphism. There exist some h ∈ G and l ∈ R d 2 such that g = (L(π(h)), l) and h · ρ = χ l ρ. Since and thus, χ g·k | T N = 1. By Lemma 4.4 we have g · k ∈ L * S /N .
is a metric space whose topology is equal to the quotient topology, see, e. g., [23, §6.6].
We are finally in a position to state and prove our main results on the structure of Ind G T F ( T F). Note that the set R in the following theorems is finite due to Corollary 4.10.

Theorem 4.16. Let R be a representation set of T F/∼. Then, the map
where is the disjoint union, is bijective.
Proof. Let R be a representation set of T F /∼. We define the map ϕ : First we show that ϕ is well-defined. Let ρ ∈ R, k, k ′ ∈ R d 2 and g ∈ G ρ such that k ′ = g · k.

Theorem 4.17. Let R be a representation set of {ρ ∈ T F | ρ|
Then the maps where is the disjoint union, Ind = Ind G T F and N ∈ M 0 in (ii) is arbitrary, are bijective. Proof. Let m ∈ N such that M 0 = mN and R be a representation set of {ρ ∈ T F | ρ| T m = I dρ }/∼. By Lemma 4.9 the set R is a representation set of T F /∼.
(i) We define the map ψ : First we show that ψ is well-defined. Let ρ ∈ R, k ∈ L * S and N ∈ M 0 . Since T N ⊂ T m and by Lemma 4.4, we have (χ k/N ρ)| T N = I dρ . By Lemma 4.13 for all N ∈ M 0 we have (L * S /N )/G ρ ⊂ R d 2 /G ρ and thus, by Theorem 4.16 the map ψ is well-defined.
As a direct consequence of the above theorems we obtain: Corollary 4.18. Let R be as in Theorem 4.16. For every σ ∈ G there exists a ρ ∈ R and a k ∈ R d 2 such that σ is a subrepresentation of Ind G T F (χ k ρ). If moreover R is as in Theorem 4.17 and ρ| T N = I dρ for an N ∈ M 0 , then k can be chosen in L * S /N . Proof. This is a direct consequence of Theorems 4.16 and 4.17 and Proposition A.7.
We finally address the natural question to what extend the induced representations labeled by ρ∈R R d 2 /G ρ from Theorem 4.16 are irreducible and exhaust the total dual space G. To this end, we equip the orbit space R d 2 /H of a d 2 -dimensional space group H with the pushforward under the action of H of the d 2 -dimensional Lebesgue-measure restricted to an associated fundamental domain. The dual space G of G shall be endowed with the Plancherel measure, cf. Theorem A.9. Without loss of generality we restrict to groups of infinite order as for |G| < ∞ one has d 2 = 0, T = {id} and T F = G.
Remark 4.20. The proof shows that the N ρ can be chosen as N ρ = G ρ · K, where K is the Lebesgue null-set Also note that, by Theorem 4.16 and Corollary 4.18, for any σ ∈ N there is a ρ ∈ R and a G ρ · k ∈ N ρ such that σ is a subrepresentation of Ind G T F (χ k ρ).

Lemma 4.21.
Let K be as in Remark 4.20. Suppose g ∈ G and k, k ′ ∈ R d 2 are such that Lemma 4.13 with ρ = χ 0 . We thus get i. e., k ′ ∈ K. If, conversely, k ′ ∈ K, then the same argument with g replaced by g −1 and s replaced by −B T s yields k ∈ K.
As a consequence of this we have the following lemma. Proof. Let k ∈ K and g ∈ G ρ , where ρ ∈ T F with ρ| T m = I dρ . There exist some h ∈ G and l ∈ R d 2 such that g = (L(π(h)), l) and h · ρ = χ l ρ. Since ρ| T m = I dρ = (h · ρ)| T m , we have χ l | T m = 1 and so l ∈ L * S /m due to Lemma 4.4. Now Lemma 4.21 implies g · k = L(π(h))k + l ∈ K.
We now prove Theorem 4.19.
Proof. As in the proof of Theorem 4.17 we let m ∈ N such that M 0 = mN and R be a representation set of {ρ ∈ T F | ρ| T m = I dρ }/∼ so that R is a also representation set of T F /∼ due to Lemma 4.9. Choose K and, for each ρ ∈ R, N ρ = G ρ · K as in Remark 4.20. We then claim that the following implication holds true for any ρ ∈ T F with ρ| T m = I dρ , k ∈ R d 2 and g ∈ G \ T F: To see this fix such ρ, k and g with g · (χ k ρ) = χ k ρ. Since ρ| T m = I dρ we also have g · χ k | T m = χ k | T m . Thus Lemmas 4.2 and 4.4 yield (L(π(g)) − I d 2 )k ∈ L * S /m and hence k ∈ K.
Since the point group L(S) is finite and the lattice L * S is countable, the Lebesgue measure of K ⊂ R d 2 is 0. As also G ρ is countable for any ρ ∈ R, the sets N ρ ⊂ R d 2 /G ρ are null-sets as well.
By Theorem 4.16 for each ρ ∈ R and G ρ · k ∈ R d 2 /G ρ there is a well-defined representation Ind G T F (χ k ρ). Combining Mackey's irreducibility criterion stated in Theorem A.8 and the above claim we see that We thus obtain that the mapping in the assertion of the theorem is well-defined and, upon choosing N ⊂ G suitably, also surjective. By Theorem 4.16 it is injective as well since, as noted above, R is a also representation set of T F/∼.
We proceed to show that N is a null-set with respect to the Plancherel measure on G. First note that, as For g ∈ G and k ∈ R d 2 Lemma 4.2 gives g · χ k | T m = χ k ′ | T m with k ′ = L(π(g))k and then Lemma 4.21 shows that k ∈ K if and only if k ′ ∈ K. Trivially, each χ k | T m extends to χ k on T F and g · χ k | T m = χ k | T m for all g ∈ T F. Conversely, if g ∈ G \ T F and k ∈ R d 2 \ K, then the above claim yields g · χ k | T m = χ k | T m . As a consequence, G χ k | T m = T F and G Invoking Theorem A.10 we thus find that is a µ G -conull subset of G, where we also have used Lemma 4.9(ii) in order to pass from representations in (T F) m to representations in T F that annihilate T m . To finish the proof it now suffices to show that Let k ∈ R d 2 \ K and ρ ∈ T F : ρ| T m = I dρ . Let ρ ′ ∈ R such that ρ ′ ∼ χ k ρ. There exists some g ∈ G and k ′ ∈ R d 2 such that g · (χ k ρ) = χ k ′ ρ ′ . We have ρ| T m = I dρ = ρ ′ | T m and thus we have χ L(π(g))k | T m = χ k ′ | T m . So with Lemma 4.4 we get L(π(g))k − k ′ ∈ L * S /m and thus k ′ ∈ R d 2 \ K due to Lemma 4.21. By Lemma 4.22 we then have G ρ · k ′ / ∈ N ρ . The claim now follows from Ind(χ k ρ) = Ind(g · (χ k ρ)) = Ind(χ k ′ ρ ′ ).

Harmonic analysis of periodic functions
In this section we develop methods from Fourier analysis for periodic mappings on G. We begin by defining a suitable notion of periodicity.
for all g ∈ G and t ∈ T N .
We equip C m×n with the Frobenius inner product · , · defined by and let · denote the induced norm. We define the set Remark 5.2. If G is finite and S a set, then every function from G to S is periodic and in particular, we have L ∞ per (G, C m×n ) = {u : G → C m×n }. The following Lemma shows that the above definition of periodicity is independent of the choice of T . Proof. Let S be a set and u : G → S be T N -periodic for some N ∈ M 0 . By Theorem 3.11 the function u is T |F|N -periodic. By Proposition 3.9 it holds and thus, we have The other direction is trivial since by Theorem 3.11 for all N ∈ N there exists some n ∈ N such that nN ∈ M 0 .
The following lemma characterizes the periodic functions on G with the aid of the quotient groups G/T N .
is well-defined. Moreover, we have and this space is a vector space.
Proof. This follows immediately from the definition of L ∞ per (G, For this choice, for all x ∈ R d and large N ∈ M 0 the set C N · x is similar to a cube which explains the nomenclature. We equip the vector space L ∞ per (G, C m×n ) with an inner product.
Definition 5.7. We define the inner product · , · on L ∞ per (G, C m×n ) by if u and v are T N -periodic for all u, v ∈ L ∞ per (G, C m×n ). We denote the induced norm by · 2 .
In order to define the Fourier transform we must choose a set of representatives for the periodic elements of G. (iii) Proposition A.4 shows that where π N is the natural surjective homomorphism from G to G N for all N ∈ M 0 . Definition 5.10. For all u ∈ L ∞ per (G, C m×n ) and for all periodic representations ρ of G we define u (ρ) : where N ∈ M 0 is such that u and ρ are T N -periodic and ⊗ denotes the Kronecker product.

Proposition 5.11 (The Plancherel formula). The Fourier transformation
is well-defined and bijective. Moreover, we have the Plancherel formula Proof. We show that the well-known Plancherel formula for finite groups, see, e. g., [26,Exercise 5.7], implies the Plancherel formula of the proposition. Let N ∈ M 0 and π N : G → G N be the natural surjective homomorphism. The map Thus the map . By the Plancherel formula for finite groups, the Fourier transformation is bijective and it holds 1 where the upper map is defined by u → (u (ρ)) ρ∈E, ρ is T N -periodic . Thus, the map is bijective and we have for all T N -periodic functions u, v ∈ L ∞ per (G, C m×n ). Since N ∈ M 0 was arbitrary, for all u ∈ L ∞ per (G, C m×n ), for all N ∈ M 0 such that u is T N -periodic and n ∈ N it holds ρ∈E, ρ is T N -periodic By (5) for all u ∈ L ∞ per (G, C m×n ) and N ∈ M 0 such that u is T N -periodic, we have By (5) and (6) the Fourier transformation · : L ∞ per (G, C m×n ) → ρ∈E C (mdρ)×(ndρ) is well-defined and we have for all u, v ∈ L ∞ per (G, C m×n ). Moreover, since the map defined in (4) is injective and the Fourier transformation is injective. Analogously, the Fourier transformation is surjective.
Remark 5.12. (i) The above proof also shows that for all u : G → C m×n and N ∈ M 0 such that u is T N -periodic, we have Moreover, for all N ∈ M 0 the map is bijective.
(ii) It is easy to see that by the above proposition we have also a description of the completion of L ∞ per (G, C m×n ) with respect to the norm · 2 . We have Lemma 5.13. Let f ∈ L ∞ per (G, C m×n ), g ∈ G and τ g f denote the translated function f ( · g). Then we have τ g f ∈ L ∞ per (G, C m×n ) and for all periodic representations ρ of G.
Proof. Let f ∈ L ∞ per (G, C m×n ), g ∈ G and ρ be a periodic representation. Let N ∈ M 0 such that f and ρ are T N -periodic. The function τ g f is T N -periodic and we have where in the third step we made a substitution and used that C N and C N g are representation sets of G/T N and that the function Definition 5.14. For all u ∈ L 1 (G, C m×n ) and all representations ρ of G we define Remark 5.15. If the group G is finite, ρ is a representation of G and u ∈ L 1 (G, C m×n ) = L ∞ per (G, C m×n ), then the Definitions 5.10 and 5.14 for u (ρ) differ by the multiplicative constant |G|, but it will always be clear from the context which of the both definitions is meant. If G is infinite, then L 1 (G, C m×n ) ∩ L ∞ per (G, C m×n ) = {0} and thus, there is no ambiguity.
Definition 5.16. For all u ∈ L 1 (G, C l×m ) and v ∈ L ∞ per (G, C m×n ) we define the convolution u * v ∈ L ∞ per (G, C l×n ) by Lemma 5.17. Let u ∈ L 1 (G, C l×m ), v ∈ L ∞ per (G, C m×n ) and ρ be a periodic representation of G. Then Proof. Let u ∈ L 1 (G, C l×m ), v ∈ L ∞ per (G, C m×n ) and ρ be a periodic representation of G. Let N ∈ M 0 such that v and ρ are T N -periodic. By Definition 5.16 it is clear that u * v is T N -periodic and thus we have u * v ∈ L ∞ per (G, C m×n ) as claimed in Definition 5.16. We

Quotient groups as semidirect products
By Definition 3.8 for all m ∈ M 0 the group T m is a normal subgroup of G, but in general there does not exist any group H < G such that G = T m ⋊ H, see Example 6.1. In this section we determine for m ∈ M 0 and appropriate N ∈ mN a group H ≤ G/T N such that see Theorem 6.7. The proof is similar to the proof of the Schur-Zassenhaus theorem, see, e. g., [1]. If G is a space group, for appropriate N ∈ N the existence of a group H such that G/T N = T /T N ⋊ H is mentioned in [3, p. 299] and in [10, p. 376]. is symmorphic and equal to T ⋊ H with T = t 1 , t 2 and H = p 1 . The space group is nonsymmorphic, since it does not contain any element of order 2, but the order of the quotient group of the space group by its subgroup of all translations is 2. For all n ∈ N coprime to |L(S)| let P (n) ⊂ G be such that the map is bijective, where π : G → S is the natural surjective homomorphism.
If n = 1, then a(n) = 0 and P Proof. Let n ∈ N be coprime to |L(S)|.
First, we prove that T n S P (n) S is a subgroup of S. Let t, s ∈ T n S and p, q ∈ P (n) We use that S∈L(S)τ (S) = S∈L(S)τ (T S) for all T ∈ L(S).
Thus, we have T n S P (n) S ≤ S. Let π be the natural surjective homomorphism from G to S with kernel F. It holds π −1 (T n S P (n) S ) = T n FP (n) and thus, T n FP (n) is a subgroup of G. Now let N ∈ (nN) ∩ M 0 . Since n divides N , we have T N ⊂ T n FP (n) . Since N ∈ M 0 , we have T N ⊳ T n FP (n) .
Recall Definition 4.8. Remark 6.5. Let n ∈ N be coprime to |L(S)|. Let m ∈ M 0 , N = nm and t 1 , . . . , t d 2 ∈ T n such that π({t 1 , . . . , t d 2 }) generates T n S , where π : T F → T S is the natural surjective homomorphism. Then, the map ((n 1 , . . . , n d 2 ), f, p) → t n 1 1 . . . t The following lemma characterizes the elements of the groups G N , (T n F) N and (T m ) N for appropriate n, m, N ∈ N. Proof. Since P (1) is a representation set of G/T F, the map T × F × P (1) → G, (t, f, p) → tf p is bijective. The assertions are clear by Lemma 3.6, Theorem 3.11 and Lemma 3.7, Proposition 3.9.
The following theorem characterizes the group G N for appropriate N ∈ N.
Theorem 6.7. Let m ∈ M 0 . Let n ∈ N be coprime to m and |L(S)|. Let N = nm. Then, we have Let n ∈ N be coprime to m and |L(S)|. Let N = nm. By Theorem 3.11 we have T m ⊳ G and T N ⊳ G, and by Lemma 3.7 we have T N ⊳ T m . Hence, we have By Lemma 3.7 the group T m is isomorphic to Z d 2 and thus, (T m ) N is isomorphic to Z d 2 n . By Lemma 6.4 we have (T n FP (n) ) N ≤ G N .
For all N ∈ N and H ≤ S such that T N S is a subgroup of H, we denote Let π : G N → S N be the natural surjective homomorphism with kernel {gT N | g ∈ F}.
We have where in the third step we used that for all p ∈ P (n) S such that L(p) = I d 2 we have p ∈ T n S and in the last step we used that the numbers n d 2 and m d 2 are coprime, |(T m S ) N | = n d 2 , |(T n S ) N | = m d 2 and Lagrange's theorem. By (9) and since π| (T m ) N is injective, we have We have see Lemma 6.6, and see Remark 6.5. By (11), (12) and since (T m ) N is isomorphic to Z d 2 n , we have By (7), (8), (10) and (13)  By Theorem 6.7 we have G N = (T m ) N ⋊ (T n FP (n) ) N . Corollary 6.9. Suppose that G is a space group. Let N ∈ N be coprime to |L(G)|. Then we have Proof. Let G be a space group. We have F = {id} and M 0 = N. For all N ∈ N coprime to |L(G)|, we have (T N P (N ) )/T N = {gT N | g ∈ P (N ) }. Thus, Theorem 6.7 implies the assertion. and Proof. This is clear by Corollary 6.8 and Corollary 6.9.
Corollary 6.11. Suppose that G = T F. Let m ∈ M 0 and n ∈ N be coprime. Let N = nm. Then it holds Proof. Suppose that G = T F. Let m ∈ M 0 and n ∈ N be coprime. We have S = T S and L(S) = {I d 2 }. Without loss of generality we assume thatτ = 0. We haveτ = 0 and P (n) S = {id}. Without loss of generality we assume that P (n) = {id}. By Theorem 6.7, Lemma 3.6(ii) and Proposition 3.9 we have G N = (T m ) N × (T n F) N .

A Dual spaces and induced representations
With the aim to keep our presentation largely self-contained and to provide easy references, we recall some topics in representation theory in a short appendix. In our set-up it is not restrictive to only consider finite-dimensional representations, see Remark A.2 below. Definition A.1. Let H be a finite group or a discrete subgroup of E(d). A representation of H is a homomorphism ρ : H → U(d ρ ), where d ρ ∈ N is the dimension of ρ and U(d ρ ) is the group of all unitary matrices in C dρ×dρ . Two representations ρ, ρ ′ of H are said to be (unitarily) equivalent if d ρ = d ρ ′ and there exists some T ∈ U(d ρ ) such that for all g ∈ H. For given representations ρ 1 , . . . , ρ n of H, we define the direct sum where d = n i=1 d ρ i . In a canonical way, the above group action and terms dimension, irreducible and direct sum are also defined for equivalence classes of representations. For any ρ ∈ N we denote by H ρ the stabilizer of ρ in H.
Remark A.2. In [21] the following theorem is proved for any locally compact group: There exists an integer M ∈ N such that the dimension of every irreducible representation is less than or equal to M if and only if there is an open abelian subgroup of finite index. This, in particular, applies to finite groups and discrete subgroups of E(d).
A caveat on notation: For a representation and for an equivalence class of representations we use the symbol χ if it is one-dimensional and ρ otherwise. For every one-dimensional representation χ its equivalence class is a singleton which we also call a representation and denote by χ.
It is elementary to see that, if H < E(d) is a discrete subgroup of E(d), then characters (one-dimensional representations) act by multiplication on H. Moreover, for any normal subgroup N of H the action of N on N is trivial: g · ρ = ρ for all g ∈ N and ρ ∈ N , i. e., H ρ ⊃ N . As a consequence, also H/N acts on N . For ρ ∈ N we denote by (H/N ) ρ the stabilizer and by (H/N ) · ρ the orbit of ρ with respect to this action. Also the following two propositions are well-known. Their proofs (in the more general setting of locally compact groups) can be found, e. g., in [15]. Our setting allows for a definition of induced representations as for finite groups, see, e. g., [26,Section 8.2]. (For general locally compact groups the definition is more complicated, see, e. g., [15,Chapter 2].) Definition A.5. Let H < E(d) be discrete and K be a subgroup of H such that the index n = |H : K| is finite. Choose a complete set of representatives {h 1 , . . . , h n } of the left cosets of K in H. Suppose ρ : K → U(d ρ ) is a representation of K. Let us introduce a dot notation in this context by settinġ The induced representation of an equivalence class of representations is the equivalence class of the induced representation of a representative. Moreover, let Ind H K ( K) denote the set of all induced representations of K. We also write Ind instead of Ind H K if K and H are clear by context.
The following proposition is standard in Clifford theory. We also record the following well-known results on the relation of irreducible and induced representations adapted to our setting. Proof. Let ρ be an irreducible representation of H. Choose σ ∈ K such that σ is a subrepresentation of ρ restricted to H. Then [18,Theorem 8.2] shows that ρ is a subrepresentation of Ind H K σ.
The following is Mackey's irreducibility criterion, cp. [17], which we state in a form that is directly implied by, e. g., [ The Haar measure (i. e., the counting measure) on a discrete group of Euclidean isometries induces a measure on the dual space as follows, cf., e. g., [11]. (If H is abelian, then µ H is (a multiple of) the Haar measure of H.) We finally state a version of the Mackey machine (cf. [19]) in a form proven in [16] and adapted to our setting of discrete subgroups of E(d).
Theorem A. 10. Let H < E(d) be discrete, N be a normal abelian subgroup of H of finite index and set P = H/N . Suppose there is an H-invariant measurable µ N -conull subset Ω ⊂ N such that all σ ∈ Ω can be extended to a unitary representationσ of H σ . For σ ∈ Ω and ρ ∈ P σ define the unitary representation σ × ρ of H σ , acting on C dσdρ , by (σ × ρ)(g) =σ(g) ⊗ ρ(gN ) for all g ∈ H σ .