Symbol functions for symmetric frameworks

We prove a variant of the well-known result that intertwiners for the bilateral shift on `$\ell^2(Z)$ are unitarily equivalent to multiplication operators on $L^2(T)$. This enables us to unify and extend fundamental aspects of rigidity theory for bar-joint frameworks with an abelian symmetry group. In particular, we formulate the symbol function for a wide class of frameworks and show how to construct generalised rigid unit modes in a variety of new contexts.

property has been utilised to obtain combinatorial characterisations of so-called forced and incidental rigidity for finite bar-joint frameworks in dimension 2. (See [12,22].) Periodic bar-joint frameworks have also received much attention in recent years. Here R(G, p) is an infinite matrix and so operator theory naturally comes to the fore. In [18], it is shown that the rigidity matrix for a periodic bar-joint framework gives rise to a Hilbert space operator which is unitarily equivalent to a multiplication operator M Φ . The symbol function Φ is matrix-valued and defined on the d-torus T d . The set of points in T d where Φ has a non-zero kernel is known as the RUM spectrum and takes its name from the phenomenon of rigid unit modes (RUMs) in silicates and zeolites (see [5,6,9]).
RUM theory for periodic bar-joint frameworks and the aforementioned decomposition theory for finite bar-joint frameworks can be viewed as two sides of the same coin. The first aim of this article is to formalise this viewpoint using techniques from Fourier analysis. The second aim is to extend the theory so that it may be applied in new contexts.
In Section 2, we prove a variant of the well-known result that intertwiners for the bilateral shift on 2 (Z) are unitarily equivalent to multiplication operators on L 2 (T) (Theorem 2.8). The distinguishing features of our theorem are that it takes place in the setting of a general locally compact abelian group, with vector-valued function spaces, and in the presence of an additional twist arising from a unitary representation.
In Section 3, we adopt the approach taken in [13] and introduce the more general notions of a framework (G, ϕ) for a pair of Hilbert spaces X and Y and an accompanying coboundary matrix C(G, ϕ). This convention simplifies the proofs and also allows the results to be applied in a much wider variety of settings (as demonstrated in the final section). Applying the results of Section 2, we show that a framework with a discrete abelian symmetry group gives rise to a Hilbert space coboundary operator C(G, ϕ) which admits a factorisation as illustrated in Figure 1 (Theorem 3.6). Note that the block diagonalisation result for finite bar-joint frameworks and the unitary equivalence result for periodic bar-joint frameworks described above both follow from this factorisation. We then provide an explicit description of the associated symbol function Φ in terms of generalised orbit matrices (Theorem 3.7) and as a trigonometric polynomial (Corollary 3.10).
In Section 4, we introduce a generalised RUM spectrum Ω(G) for frameworks with a discrete abelian symmetry group Γ and show how to construct χ-symmetric vectors z(χ, a) which lie in the kernel of the coboundary matrix C(G, ϕ) for each χ ∈ Ω(G) (Theorem 4.1). Note that here we continue to work in the more general setting of coboundary operators and that the RUM spectrum is presented as a subset of the dual groupΓ. In the terminology of [5,6,9], characters χ ∈Γ correspond to wave-vectors in reciprocal space and χ-symmetric vectors which lie in the kernel of C(G, ϕ) correspond to generalised rigid unit modes.
Finally, in Section 5, we illustrate the results of the preceding sections with several contrasting examples. These include a bar-joint framework in R 3 with screw axis symmetry, a direction-length framework in R 2 with both translational and reflectional symmetry and a symmetric bar-joint framework in R 3 with mixed-norm distance constraints. For each example, we provide some necessary background, formulate the symbol function Φ, compute the RUM spectrum Ω(G) and construct generalised rigid unit modes z(χ, a) for points χ ∈ Ω(G). To the best of our knowledge, the interplay between rigidity and symmetry has not previously been explored in these contexts.

Intertwining relations
Let Γ be a locally compact Hausdorff abelian group. Denote by L 2 (Γ) the Hilbert space of square integrable functions, i.e. Borel-measurable functions f : Γ → C such that, where we use normalised Haar measure on Γ. Recall the Haar measure of a locally compact group is decomposable on Γ; in particular, Γ contains a σ-compact clopen subgroup ( [7]). If S is a selfadjoint set, i.e. S * ∈ S for all S ∈ S, then S is also selfadjoint and hence a C * -algebra. Moreover, S is a set of commuting operators if and only if S ⊆ S . Thus, an operator set is maximal abelian if and only if S = S ( [16]).
Proof. M µ is abelian, so M µ is a subset of its commutant. For the reverse inclusion, let T ∈ (M µ ) . We shall show that there exists g ∈ L ∞ (Γ), such that T = M g .
(i) Suppose first that Γ is compact, so µ(Γ) < ∞. Then the constant function 1 Γ lies in L 2 (Γ). Define g = T 1 Γ ∈ L 2 (Γ). Then for every f ∈ L ∞ (Γ), we have Hence, it suffices to show that g ∈ L ∞ (Γ). Let α > 0 and Γ α = {γ ∈ Γ : |g(γ)| > α}. Let 1 α be the characteristic function of Γ α . Then hence α ≤ T whenever µ(Γ α ) > 0. Thus g ∞ ≤ T . (ii) Suppose now that Γ is σ-compact. Then Γ can be written as a countable union of pairwise disjoint precompact sets Γ n . Write 1 n for the characteristic function of Γ n and let g n = T 1 n . Similarly to the previous case, we obtain that T M 1n = M gn and g n ∞ ≤ T for every n ∈ N. Hence define g ∈ L ∞ (Γ) by g Γn = g n Γn , for every n ∈ N. Then g ∞ ≤ sup n g n ∞ ≤ T , so g ∈ L ∞ (Γ), and for every f ∈ L 2 (Γ) we have (Each of the infinite sums should be interpreted as limits in L 2 of the partial sums.) (iii) In the general case, let H be a clopen σ-compact subgroup of Γ and let Z be a subset of Γ that contains exactly one element of each coset of H, so that Γ can be written as the disjoint union of the sets z+H, z ∈ Z. For each z ∈ Z, denote by 1 z the characteristic function of z+H and let g z = T 1 z . Similarly to the above cases, we have T M 1z = M gz and g z ∞ ≤ T for every z ∈ Z. Define g ∈ L ∞ (Γ) by g z+H = g z z+H , for every z ∈ Z. Then g is locally almost everywhere well-defined, g ∞ ≤ sup z g z ∞ ≤ T , so g ∈ L ∞ (Γ). Now given any function f ∈ L 2 (Γ), there exists a countable family {z n : n ∈ N} ⊆ Z such that the set supp(f ) ∩ (Γ\(∪ n z n + H)) is null ([21, Appendix E8]). Check that since T commutes with the multiplication operators of characteristic functions, it follows that supp(T f ) ⊆ supp(f ). Hence The Fourier transform F : (L 1 ∩ L 2 )(Γ) → L 2 (Γ) given by the formulâ extends uniquely to a unitary isomorphism from L 2 (Γ) to L 2 (Γ) ( [7,21]). The inverse Fourier transform of a function f ∈ L 2 (Γ) is denotedf . For each γ ∈ Γ, denote by D γ the unitary operator Also, denote by δ γ ∈Γ, the scalar function δ γ (ξ) = ξ(γ) for each ξ ∈Γ. Note that the map δ : Γ →Γ, γ → δ γ , is the Pontryagin map ( [7]).
The result now follows by Proposition 2.2.
Proposition 2.4. Let L ∈ B(L 2 (Γ)). Then L satisfies the commuting property D γ L = LD γ for all γ ∈ Γ if and only if L is unitarily equivalent to a multiplication operator M Φ ∈ B(L 2 (Γ)) for some Φ ∈ L ∞ (Γ). In particular, Proof. Suppose first that L ∈ B(L 2 (Γ)) and D γ L = LD γ for all γ ∈ Γ. By Corollary 2.3, setting Λ = F LF −1 ∈ B(L 2 (Γ)), we obtain that Similarly, for all γ ∈ Γ, Therefore, by the uniqueness of the Fourier transform we obtain It now follows that, for all h ∈ L ∞ (Γ), for every f, g ∈ L 2 (Γ) ∩ L ∞ (Γ), and since these functions are dense in L 2 , we get M h Λ = ΛM h , so Λ commutes with the algebra M µ of multiplication operators. Thus, the result follows from Proposition 2.1.
The reverse direction is obtained from Corollary 2.3, so the proof is complete.
Remark 2.5. If Γ is a discrete abelian group and Φ ∈ L 1 (Γ) then the operator L in Proposition 2.4 satisfies, for all γ ∈ Γ. In particular, if Γ = Z then the matrix for L is the Laurent matrix with symbol Φ.

2.2.
Vector-valued functions. Let Γ be a locally compact abelian group and let X and Y be complex Hilbert spaces. Let also {x 1 , x 2 , . . . } and {y 1 , y 2 , . . . } be orthonormal bases on X and Y , respectively. Denote by L 2 (Γ, X) the Hilbert space of square integrable X-valued functions. i.e. Bochner-measurable functions f : Γ → X such that, where we use normalised Haar measure on Γ. Note that we identify the Hilbert spaces L 2 (Γ, X) and L 2 (Γ) ⊗ X; given any g ∈ L 2 (Γ), the function g k ∈ L 2 (Γ, X) defined by The Fourier transform F X ∈ B(L 2 (Γ, X), L 2 (Γ, X)) is the unitary operator given by F X = F ⊗ 1 X , where 1 X is the identity operator on X. For each γ ∈ Γ, denote by U γ and W γ the unitary operators Given now an operator T ∈ B(L 2 (Γ, X), L 2 (Γ, Y )), for each i, j let T ij ∈ B(L 2 (Γ)) be the bounded operator that is uniquely defined by the sesquilinear form, We call T ij a matrix element of T . A bounded operator T ∈ B(L 2 (Γ, X), L 2 (Γ, Y )) is called a multiplication operator if there exists Φ ∈ L ∞ (Γ, B(X, Y )) such that We refer to the function Φ as the operator-valued symbol function for T and we write T = M Φ . In terms of the matrix elements T ij from (1), Suppose that the intertwining property holds. Then for every f, g ∈ L 2 (Γ) we have X . This is a bounded operator that satisfies Once again, the reverse direction follows by straightforward calculations.

2.3.
Intertwining with a twist. Let U (X) denote the unitary group of X and let π : Lemma 2.7. Let π : Γ → U (X) be a unitary representation. Then, for each γ ∈ Γ, Proof. Given f ∈ L 2 (Γ, X) and γ ∈ Γ, we have , so the proof is complete.
The conclusion now follows from Proposition 2.6 on taking L = CT −1 π . Conversely, suppose C = LT π , where L is unitarily equivalent to a multiplication operator ). By Proposition 2.6 and Lemma 2.7, for each γ ∈ Γ,

Symbol functions for symmetric frameworks
In this section we introduce frameworks (G, ϕ) and their associated coboundary matrices C(G, ϕ). We show that the action of a discrete abelian group on (G, ϕ) gives rise to a Hilbert space coboundary operator which satisfies twisted intertwining relations of the form considered in Section 2. In particular, this coboundary operator can be expressed as a composition LT π in the manner of Theorem 2.8, where L is unitarily equivalent to a multiplication operator M Φ . We then present an explicit formula for the operator-valued symbol function Φ.
3.1. Frameworks. Let X and Y be finite dimensional complex Hilbert spaces. A framework for X and Y is a pair (G, ϕ) consisting of a simple undirected graph G = (V, E) and a collection ϕ = (ϕ v,w ) v,w∈V of linear maps ϕ v,w : X → Y with the property that ϕ v,w = 0 if vw / ∈ E and ϕ v,w = −ϕ w,v for all vw ∈ E. We will assume throughout this section that the vertex set V is a finite or countably infinite set. The graph G is said to have bounded A coboundary matrix for (G, ϕ) is a matrix C(G, ϕ) with rows indexed by E and columns indexed by V . The row entries for a given edge vw ∈ E are as follows, A coboundary matrix for (G, ϕ) has the following form (up to permutations of rows and columns), Note that a coboundary matrix gives rise to the linear map, We recall the following result. . Let (G, ϕ) be a framework for X and Y . If G is a countably infinite graph with bounded degree then the following statements are equivalent. Figure 2. A 4-cycle (left) and coboundary matrix (right).
3.2. Gain graphs. Let Γ be an additive group with identity element 0. A Γ-symmetric graph is a pair (G, θ) where G = (V, E) is a simple undirected graph with automorphism group Aut(G) and θ : Γ → Aut(G) is a group homomorphism. For convenience, we suppress θ and write γv instead of θ(γ)v for each group element γ ∈ Γ and each vertex v ∈ V . We also write γe instead of (γv)(γw) for each γ ∈ Γ and each edge e = vw ∈ E. The orbit of a vertex v ∈ V (respectively, an edge e ∈ E) under θ is the set [v] = {γv : γ ∈ Γ} (respectively, [e] = {γe : γ ∈ Γ}). We denote by V 0 the set of all vertex orbits and by E 0 the set of all edge orbits. We will assume throughout that θ acts freely on the vertices and edges of G. This means γv = v and γe = e for all γ ∈ Γ\{0} and for all vertices v ∈ V and edges e ∈ E. We will also assume that V 0 and E 0 are finite sets. Lemma 3.3. Let (G, θ) be a Γ-symmetric graph where θ acts freely on the vertices and edges of G and E 0 is finite. Then G has bounded degree.
The quotient graph G 0 is the multigraph with vertex set V 0 , edge set E 0 and incidence relation satisfying and is denoted ψ [e] . A gain graph for the Γ-symmetric graph (G, θ) is any edge-labelled directed multigraph obtained from the quotient graph G 0 in this way. Figure 3. . The set of all such representative edges will be denotedẼ 0 . Note that since θ acts freely on the vertex set V and edge set E we have natural bijections, For more on gain graphs we refer the reader to [12].
3.3. Symmetric frameworks. Let Γ be a discrete abelian group and denote by Isom(X) the group of affine isometries of X. A Γ-symmetric framework is a tuple G = (G, ϕ, θ, τ ) where τ : Γ → Isom(X) is a group homomorphism, (G, θ) is a Γ-symmetric graph and (G, ϕ) is a framework for X and Y with the property that, ϕ γv,γw = ϕ v,w • τ (−γ), for all γ ∈ Γ and all v, w ∈ V.
For each γ ∈ Γ, let dτ (γ) denote the linear isometry on X that is uniquely defined by the linear part of the affine isometry τ (γ). We denote byτ : Given a vector z = (z v ) v∈V ∈ X V we will write z e = z v − z w for each edge e = vw ∈ E where the corresponding directed edge [e] in the gain graph is directed from [v] to [w]. We will also write ϕ e = ϕ v,w for such an edge.
We refer to Φ in the above theorem as the symbol function for the symmetric framework G.
Note that each orbit matrix gives rise in natural way to a linear map O G (χ) : We now show that O G is the symbol function for the symmetric framework G.
Let [e] = ([v], [w]) ∈ E 0 be a directed edge with gain γ ∈ Γ and let g [e] ∈ 2 (Γ, Y ) be the [e]-component of g. Note that for each γ ∈ Γ, Also, by Proposition 2.2, for almost every χ ∈Γ, and so, Thus, for almost every χ ∈Γ, Corollary 3.8. Let G = (G, ϕ, θ, τ ) be a Γ-symmetric framework with symbol function Φ. If G is a finite graph then the coboundary matrix C(G, ϕ) is equivalent to the direct sum, Proof. By Theorem 3.6, C(G, ϕ) is equivalent to M Φ . Note that since G is a finite graph and θ acts freely on the vertices and edges of G it follows that Γ, and hence alsoΓ, is finite. Thus, M Φ is equivalent to the direct sum ⊕ χ∈Γ Φ(χ). Also, by Theorem 3.7, Φ(χ) = O G (χ) for all χ ∈Γ and so the result follows. If G = (G, ϕ, θ, τ ) is a Z 2 -symmetric framework then the associated orbit matrices for G take the following form, Applying Corollary 3.8 we obtain the equivalence, Corollary 3.10. Let G = (G, ϕ, θ, τ ) be a Γ-symmetric framework with symbol function Φ = O G ∈ C(Γ, B(X V 0 , Y E 0 )). Fix a gain graph for (G, θ) and let Γ 0 ⊂ Γ be the finite set of non-zero gains on the edges of this gain graph.
Remark 3.11. The orbit matrix O G (1Γ) was first introduced in [23] in the context of finite bar-joint frameworks (G, p) with an abelian symmetry group. There the linear maps ϕ v,w are derived from Euclidean distance constraints and the orbit matrix is used to analyze fully symmetric motions of the framework in Euclidean space R d . The general orbit matrices O G (χ) were later introduced in [22] and used to derive the block-diagonalisation result in Corollary 3.8. The symbol function Φ for periodic bar-joint frameworks in R d , again with Euclidean distance constraints, was first introduced in [18]. In this setting the symmetry group is Z d and the dual group is the d-torus T d . It is proved there that the rigidity matrix for the framework determines a Hilbert space operator R(G, p) : 2 (V, C d ) → 2 (E, C) which is unitarily equivalent to the multiplication operator M Φ : Theorem 3.7 unifies and generalises these two contexts to frameworks with a general (finite or infinite) discrete abelian symmetry group and arbitrary linear edge constraints. See Section 5 for some examples.

A generalised RUM spectrum
Let G = (G, ϕ, θ, τ ) be a Γ-symmetric framework for X and Y with symbol function Φ ∈ C(Γ, B(X V 0 , Y E 0 )). Fix χ ∈Γ and a ∈ X V 0 and define z(χ, a) = (z v ) v∈V ∈ ∞ (V, X) to be the bounded vector with components, We refer to z(χ, a) as a χ-symmetric vector in ∞ (V, X).
In this section our aim is to prove the following result.
Proof. By Lemmas 4.3 and 4.4 we have, ν λ w * → 0 and ν λ w * → ρ(χ, a). Since the w * -topology is Hausdorff it follows that ρ(χ, a) = 0. Thus the function f χ,a ∈ ∞ (Γ, X V 0 ) given by, The Rigid Unit Mode (RUM) spectrum of G is defined as follows, Remark 4.5. The study of rigid unit modes and the RUM spectrum was initiated in [9] as a means of understanding phase-transitions and structural stability in minerals. An operator-theoretic formulation of these notions was introduced by Owen and Power in the context of periodic bar-joint frameworks in Euclidean space R d ( [18]). In the above generalisation, characters χ in the dual groupΓ can be thought of as wave vectors in reciprocal space. The χ-symmetric vectors z(χ, a) which lie in the kernel of C(G, ϕ) correspond to generalised rigid unit modes for the symmetric framework.

Examples from discrete geometry
In this section we present some contrasting examples of symmetric frameworks arising from systems of geometric constraints. In each case, the underlying geometric structure is provided by a simple undirected graph G, a normed linear space X and an assignment p : V → X of points in X to each vertex in G. We consider 1) Euclidean distance constraints for a bar-joint framework with screw axis symmetry, 2) a direction-length framework with both periodic and reflectional symmetry and 3) mixed-norm distance contraints for a finite bar-joint framework with symmetry group C 4h . Each vector in the kernel of the associated coboundary matrix C(G, ϕ) represents an infinitesimal (or first-order) flex of the framework. We derive the symbol function Φ, compute the RUM spectrum Ω(G) and construct χ-symmetric infinitesimal flexes (i.e. generalised rigid unit modes) for these frameworks.

5.1.
Bar-joint frameworks in R d . A bar-joint framework in R d is a pair (G, p) consisting of a simple undirected graph G = (V, E) and a point p = (p v ) v∈V ∈ (R d ) V with the property that p v = p w whenever vw ∈ E. For each pair v, w ∈ V , set ϕ v,w : vw ∈ E and ϕ v,w = 0 otherwise. Then the pair (G, ϕ) is a framework (for the Hilbert spaces C d and C) in the sense of Section 3.
Expressing each linear map ϕ v,w as a row vector we obtain the rigidity matrix R(G, p) with rows indexed by E and columns indexed by V × {1, . . . , d}. The row entries for a given edge vw ∈ E are as follows, We begin with a small example.
The bar-joint framework (G, p) is illustrated in Figure 2 together with an accompanying rigidity matrix R(G, p). Let θ : Z 2 → Aut(G) be the group homomorphism described in Example 3.4. Let τ : Z 2 → Isom(R 2 ) be the group homomorphism for which τ (1) is the orthogonal reflection in the line y = 1 2 . Then G = (G, ϕ, θ, τ ) is a Z 2 -symmetric framework. With the notation of Example 3.9, the symbol function for G satisfies, The multiplication operator M Φ takes the form x y .
In particular, we obtain the block diagonalisation of the rigidity matrix R(G, p) noted in Corollary 3.8, Note that Ω(G) = {χ 0 , χ 1 }. The χ 0 -symmetric infinitesimal flexes derive from fully symmetric motions of the framework and take the form, , where a, b ∈ C. The χ 1 -symmetric infinitesimal flexes take the form, We now present our first main example.
Consider the Z-symmetric framework G dh = (G dh , ϕ, θ, τ ). To formulate the symbol function for G dh we first compute, Recall that the dual group of Z consists of characters of the form χ ω : where ω ∈ T. Thus, by Theorem 3.7, the symbol function Φ : T → M 3×6 (C) is given by, Note that Φ(ω) has a 3-dimensional kernel for all ω ∈ T and so Ω(G dh ) = T.

Direction-length frameworks.
A direction-length framework in R d is a pair (G, p) consisting of a simple undirected graph G = (V, E), a partition of the edge set E into two subsets D and L, and a point p = (p v ) v∈V ∈ (R d ) V with the property that p v = p w whenever vw ∈ E. For each pair v, w ∈ V , set ϕ v,w : C d → C d−1 to be, (i) a linear map with rank d − 1 and kernel spanned by p v − p w , if vw ∈ D, (ii) the linear map x → ((p v − p w ) · x)I d−1 , if vw ∈ L, and, (iii) 0, if vw / ∈ E. Note that the pair (G, ϕ) is a framework (for the Hilbert spaces C d and C d−1 ) in the sense of Section 3. The edges in D represent direction constraints and the edges in L represent length constraints. Mixed constraint systems of this type arise naturally in CAD and network localisation for example (see [24,11]). Example 5.3. (Diamond lattice framework) Consider the diamond lattice direction-length framework illustrated in Figure 6. The graph G dl has vertex set V = {v n,j : n ∈ Z, j ∈ {0, 1}} and edge set E = D ∪ L where D = {v n,j v n+1,j : n ∈ Z, j ∈ {0, 1}} and L = {v n,0 v n+1,1 , v n,0 v n−1,1 : n ∈ Z, j ∈ {0, 1}}. The placement p of G dl in R 2 satisfies p n,j := p(v n,j ) = (n, (−1) j+1 ) for all n ∈ Z and j ∈ Z 2 .
Given v, w ∈ V , define ϕ v,w : C 2 → C by setting, Then (G, ϕ) is a framework (for the Hilbert spaces C 2 and C) in the sense of Section 3.
Define a group homomorphism θ : Then the pair (G dl , θ) is a Z × Z 2 -symmetric graph. The accompanying gain graph [v 0,0 ] Figure 6. The diamond lattice direction-length framework G dl (left) and its gain graph (right).
Define a group homomorphism τ : Z × Z 2 → Isom(R 2 ) with linear part, dτ (m, j) = 1 0 0 (−1) j , m ∈ Z, j ∈ Z 2 . and translation vector ( 1 0 ). Note that θ and τ satisfy, The isometry group Isom(X) is a subgroup of Isom(X ) (see [25,Corollary 3.3.4]) and each isometry of X has a natural extension to an isometry of X C . Thus, regarding τ as a homomorphism into Isom(X C ), we see that G = (G, ϕ, θ, τ ) is a Γ-symmetric framework in the sense of Section 3.
Example 5.4. ( 3 2,q distance constraints) Let 3 2,q , where q ∈ (1, ∞), denote the vector space R 3 equipped with the smooth mixed (2, q)-norm in R 3 given by, (x, y, z) 2,q = ((x 2 + y 2 ) q 2 + |z| q ) 1 q . Infinitesimal rigidity for non-symmetric bar-joint frameworks in these spaces has recently been studied in [3]. In particular, it is shown there that the Lowner ellipsoid for the unit ball in 3 2,q is the Euclidean unit ball in R 3 . Thus the associated complex Hilbert space is C 3 .
Then the pair (G bk , θ) is a Z 4 × Z 2 -symmetric graph. The accompanying gain graph Note that, p v m+n,j+k = τ (m, j)p n,k , ∀ m, n ∈ Z 4 , j, k ∈ Z 2 .