Generic spectrum of the weighted Laplacian operator on Cayley graphs
Cristian F. Coletti, Lucas R. de Lima, Diego S. de Oliveira, Marcus A. M. Marrocos
aa r X i v : . [ m a t h . SP ] S e p GENERIC SPECTRUM OF THE WEIGHTED LAPLACIANOPERATOR ON CAYLEY GRAPHS
CRISTIAN F. COLETTI, LUCAS R. DE LIMA, DIEGO S. DE OLIVEIRA,AND MARCUS A. M. MARROCOS
Abstract.
In this paper we address the problem of determining whether theeigenspaces of a class of weighted Laplacians on Cayley graphs are genericallyirreducible or not. This work is divided into two parts. In the first part,we express the weighted Laplacian on Cayley graphs as the divergence ofa gradient in an analogous way to the approach adopted in Riemanniangeometry. In the second part, we analyze its spectrum on left-invariant Cayleygraphs endowed with an invariant metric in both directed and undirected cases.We give some criteria for a given eigenspace being generically irreducible.Finally, we introduce an additional operator which is comparable to theLaplacian, and we verify that the same criteria hold.
Contents
1. Introduction 12. Preliminaries 32.1. Basic definitions 32.2. Differential calculus on graphs 32.3. Laplacian on graphs 52.4. Differential structure of Cayley graphs 73. Main results 73.1. Schueth’s criteria for finite Cayley graphs 94. Applications of Theorem 3.5 115. Criterion for undirected Cayley graphs 146. The operator L w Introduction
Spectra of graphs have received much attention since its foundations were laid inthe fifties and sixties of the 20 th century. The reason for that is a wealth of appliedproblems that can be posed as spectra problems on graphs. Mathematics Subject Classification.
Key words and phrases.
Weighted Laplacian, Eigenvalues, Cayley graphs.The first author was supported in part by grant
In this work, the spectra of a graph is reffered to as the set of eigenvalues ofa “Laplacian” operator. However, there exists no canonical notion of Laplacianoperator on graphs, although some of them are quite common. In this work wedeal with a weighted Laplacian on graphs, see § C k -Riemannian metrics. Recently, many workshave been addressed to this question on the setting of graphs, see [1, 2, 7, 11].In the presence of symmetry (the operator commutes with a group G of unitaryoperators) we can not expect simplicity of eigenvalues of the Laplacian since the realeigenspaces are orthogonal representations of the group G . This situation occur,for instance, when in a Riemannian Manifold one restricts to metrics such that theisometry group contains a group G fixed. Therefore, it can be conjectured thatfor a generic metric all real eigenspaces are irreducible representations of G . Thisquestion was addressed in many works in the literature, see [1, 2, 9, 10, 16].Zelditch [16] established the generic situation of spectrum of the Laplacian inthe G -manifolds with a finite group G . The higher symmetry case occurs whenthe group G acts transitively on the manifold, see [13, 10]. Schueth [13] provideda completely algebraic criteria for the existence of left invariant metrics compactLie group G such that each eigenspace of the Laplacian operator has an irreduciblerepresentation under the action of G .In this paper our goal is to establish the generic situation of the spectrum of theLaplacian on finite Cayley graphs having the set of weights as parameter space. Inorder to do that we adapt Schueth’s method to Cayley graphs, see Theorem 3.5.We apply the criteria obtained to several examples. In our knowledge, up to now,Berkolaiko and Liu [2] is the only work in the setting of graphs with symmetries.They found a family of G -invariant combinatorial Laplacians on graph were theirreducibility of the eigenspace does not occur when G is the tetrahedron symmetrygroup. We point out that we provided families of graphs where the conjecture holdsand does not hold.We present the Laplacian theory on graphs keeping in mind the intrinsicdefinition of the Laplacian operator on Riemannian manifold. Concepts such astangent space, metric, connections, differential operators are defined on graphs,in particular the Laplacian operator takes the same form that in Riemanniangeometry. When we specialized this approach to Cayley graphs we can analyzethe invariant operators similarly to compact Lie group with positive dimension. Anaive reason to do that is to clarify how the Schueth’s method must work in theCayley graph setting. A more ambitious reason is to use this approach to bring asmuch as possible classical results on continuous case to discrete one. We hope toexplore this analogy deeply in future works.This paper is organized as follows. In section 2, we present the theory of thelaplacians on graphs. This section follows the same spirit of [5, 6]. In section 3,we prove the Schueth’s criteria for Cayley graphs, Theorem 3.5. In section 4 weexhibit some examples of Cayley graphs where we apply the criteria. In section 5we deal with undirected graphs and we establish a Schueth’s criteria, and give someapplications. Finally, in section 6 we define the operator L w which is analogous tothe Laplacian, and we verify that similar results can be drawn. ENERIC SPECTRUM OF THE WEIGHTED LAPLACIAN ON FINITE GROUPS 3 Preliminaries
In this section, we define the essential elements to be studied. It contains asummary of topics on first-order differential calculus on graphs. We define in thisstructure vector fields and the notion its derivatives given by the construction of aconnection. The Laplacian operator is defined later.2.1.
Basic definitions.
We begin by setting up the notation and terminology forsome elements of graph theory.A graph G = ( V, E ) is said to be a directed graph (or digraph) if the vertexset V is non-empty and E ⊆ V × V . The elements of E are called directed edges.Let x y or y < x denote the existence of the directed edge ( x, y ) ∈ E beginningat x and ending at y . We shall write x ∼ y for the case when x y and x < y .An undirected graph G = ( V, E ) differs from a digraph by the set E whereeach of its elements (edges) may be represented by { x, y } with x, y ∈ V , insteadof an ordered pair. However, given an undirected graph G = ( V, E ) there exists anequivalent associated directed graph G = ( V, E ) which ( x, y ) ∈ E if, and only if, { x, y } ∈ E ; hence x ∼ y . Therefore, once the results are shown for directed graphs,they can be extended to undirected graphs under some required adjustments.A graph automorphism is a bijection F : V → V such that, for each directededge ( x, y ) ∈ E , one has F ( y ) < ϕ ( x ).The weight of a directed edge ( x, y ) ∈ E is given by a function w : E → R ∗ + .For simplicity of notation we write w ( x, y ). If it is associated to an undirectedgraph, then w is assumed to be symmetric. The ordered pair ( G , w ) can be seenas a length space and it will be associated with a differential structure in the nextsubsection.We give below the definition of Cayley graphs. They define a class of graphswith particular properties of our interest in the study of the Laplacian operators,as disclosed in the next sections. Definition (Cayley graph) . Let (
G, . ) be a group and let S ⊆ G . The left-invariantdirected Cayley graph C ( G, S ) = (
V, E ) is defined by V = G and E = (cid:8) ( x, xs ) : x ∈ G, s ∈ S (cid:9) . Throughout this paper, S will be considered to be a finite generator of G suchthat e S , where e is the neutral element of the group G . We may considerconstant weight along the edges defined by a s ∈ S . In this case fix w s > s ∈ S and set w ( x, xs ) = w s for all x ∈ G .2.2. Differential calculus on graphs.
For the convenience of the reader, we givebelow the construction of differential calculus over an algebra of functions. Thedefinitions are adapted to our purposes and roughly more general than the onespresented in [5]. The idea is to establish an analogy between some concepts anthose of Riemannian geometry (see for instance [4, Ch. 1]).Let G = ( V, E ) be a locally finite discrete digraph, i.e. , the set of out-neighbors N + ( x ) := { y ∈ V : y < x } is finite for every x ∈ V and V is a discrete set. Denoteby A K the K -algebra of the functions f : V → K , where K is the field R or C ofreal or complex numbers. We shall denote A K by A when K does not require tobe specified. Observe that every f ∈ A can be written as f = P x ∈ V f ( x ) δ x with δ x ( y ) = δ xy . Here δ stands for the Kronecker delta. By abuse of notation, f is C. F. COLETTI, L. R. DE LIMA, D. S. DE OLIVEIRA, AND M. A. M. MARROCOS written as the given sum even when the graph is infinite. It is justified and will beused again in what follows due to the linearity relations and local finiteness.The derivative of a function f ∈ A on an edge ( x, y ) ∈ E can be defined by thediscrete derivative ∂ yx ( f ) = (cid:0) f ( y ) − f ( x ) (cid:1) δ x . (2.1)Associate an algebraic object ð yx with each edge ( x, y ) ∈ E . We define the tangent space at x as the free module over K given by T x V = span K { ð yx : y < x } . Let us denote by
T V the tangent bundle , which is defined as the union F x ∈ V T x V .We write df ( x ) : T x V → K for the differential of f at x as the operator given bylinear extension of df ( x ) . ð yx = ∂ yx ( f )( x ) . Let dF ( x ) : T x V → T F ( x ) V denote the differential of a graph automorphim F at x given by linear extension of dF ( x ) (cid:0) ð yz (cid:1) = δ z ( x ) ð F ( y ) F ( z ) . (2.2)Let y < x , then we can define, for each z ∈ V , ∂ yx ( z ) = δ x ( z ) ð yx . (2.3)A map X : V → T V is called a vector field when for each x ∈ V we have X ( x ) ∈ T x V written as X ( x ) = X y < x X ( x ) y ð yx with X ( x ) y ∈ K . (2.4)We will denote by X the space of vector fields. Let X, Y ∈ X , α ∈ K , and f ∈ A .Then we can naturally define( X + Y )( x ) := X ( x ) + Y ( x ) , ( αX )( x ) := αX ( x ) , and ( f X )( x ) := f ( x ) X ( x ) , ∀ x ∈ V. After considering the properties above, (2.3), and (2.4), we may write X ∈ X asthe sum X = X y < x X y ∂ yx , where the functions X y ∈ A are given by X y ( x ) = X ( x ) y for each x, y ∈ V . Itfollows from (2.1) that every X ∈ X becomes a linear operator X : A → A suchthat X ( f ) = X y < x X y ∂ yx ( f ) . Now, for each weight w , we can construct an inner product g w ( x ) on the vectorspace T x V by setting g w ( x ) ( w ( x, y ) / ð yx , w ( x, z ) / ð zx ) := δ zx . Equivalently, g w ( x ) ( ð yx , ð zx ) = w ( x, y ) − / w ( x, z ) − / δ zx . ENERIC SPECTRUM OF THE WEIGHTED LAPLACIAN ON FINITE GROUPS 5
The application g w that assigns the inner product g ( x ) to each T x V is called metric tensor induced by the weight w . One can easily see g w as the metric thatsets out { w ( x, y ) / ð yx ; y < x } as a orthonormal basis of T x V for each x ∈ V .Set X x := span K { ∂ yx ∈ X : y < x } for a given x ∈ V . Observe that X x and T x V are canonically isomorphic. Then g w can also be seen as an application g w : X × X −→ A given by g w ( X, Y )( x ) := g w ( x ) ( X ( x ) , Y ( x )) . (2.5)The gradient of a function f ∈ A associated with g w is the vector field denotedby grad f such that, for each X ∈ X , X ( f ) = g w ( X, grad f ) . Note that the gradient is uniquely defined by using the Riesz Theorem on X x forevery x ∈ V .2.3. Laplacian on graphs.
We aim to define the Laplacian operator formally.Then it is desirable to establish a notion of derivative of vector fields to define thedivergence operator similarly as done in Riemannian geometry by affine connections(namely the Levi-Civita connection).The derivative (2.1) satisfies the following product rule: ∂ yx ( f g ) = ∂ yx ( f ) g + f ∂ yx ( g ) + ∂ yx ( f ) ∂ yx ( g ) . (P)The identity above can be seen as a modified Leibniz rule. It inspires us to findan application ∇ : X × X −→ X such that ∇ ( X, Y )
7→ ∇ X Y with the followingproperties: ∇ f∂ yx + ∂ qp Y = f ∇ ∂ yx Y + ∇ ∂ qp Y, (2.6) ∇ ∂ yx ( Y + Y ) = ∇ ∂ yx Y + ∇ ∂ yx Y , (2.7) ∇ ∂ yx ( f Y ) = ∂ yx ( f ) Y + f ∇ ∂ yx Y + ∂ yx ( f ) ∇ ∂ yx Y, (2.8) ∂ yx (cid:0) g w ( ∂ qp , ∂ vx ) (cid:1) = g w ( ∇ ∂ yx ∂ qp , ∂ vx ) + g w ( ∂ qp , ∇ ∂ yx ∂ vx )+ g w ( ∇ ∂ yx ∂ qp , ∇ ∂ yx ∂ vx ) . (2.9)Equations (2.8) and (2.9) are analogous to the usual Leibniz rule and metriccompatibility found in Riemannian geometry adapted to the the property (P).The so obtained map ∇ satisfying (2.6), (2.7), (2.8) and (2.9) will be called a w -connection of G associated with the weight w (or with the metric g w ). Thevector field ∇ X Y is said to be, roughly speaking, the derivative of Y along X .Let ( x, y ) , ( p, q ) ∈ E and define ∇ by bilinear extension of ∇ ∂ yx ∂ qp = δ x δ p (cid:0) ∂ qp ( y ) − ∂ qp ( x ) (cid:1) , (2.10) i.e. , the vector field ∇ ∂ yx ∂ qp is such that z δ x ( z ) δ p ( z ) (cid:0) ∂ qp ( y ) − ∂ qp ( x ) (cid:1) for each z ∈ V . For simplicity of notation, we shall write ∇ yx X instead of ∇ ∂ yx X . Proposition 2.1.
The map ∇ given by (2.10) is a well-defined w -connectionsatisfying ∇ ∂ yx ∂ qp = − δ x ∂ qp . (2.11) Proof.
First note that the properties (2.6) and (2.7) are immediate consequencesof the bilinear extension of (2.10). It also follows that ∇ ∂ yx Y = − P z < x Y ( x ) y ∂ zx ,which in particular proves (2.11). C. F. COLETTI, L. R. DE LIMA, D. S. DE OLIVEIRA, AND M. A. M. MARROCOS
Futhermore, we have ∂ yx ( f ) Y = P z < x ∂ yx ( f ) Y ( x ) z ∂ zx = − ∂ yx ( f ) ∇ ∂ yx Y . Theproperty (2.8) is obtained by observing that ∇ ∂ yx f Y = − P z < x f ( x ) Y ( x ) z = f ∇ ∂ yx Y .It remains to check (2.9). Applying the results obtained above, it is clearthat g w ( ∂ qp + ∇ ∂ yx ∂ qp , ∇ ∂ yx ∂ vx ) = 0. We see at once that ∂ yx (cid:0) g w ( ∂ qp , ∂ vu ) (cid:1) = − g w ( ∂ qp ( x ) , ∂ vx ( x )) δ x = g w ( ∂ qp ( x ) , ∇ ∂ yx ∂ vx ( x )) δ x , and the proof is complete. (cid:3) Let x ∈ V and fix the basis { ð yx : y < x } . Then we can define ð yx
7→ ∇ yx X ( x )by linear extension. Now we establish the divergence of a vector field X as theoperator div : X → A given by(div X )( x ) := trace { ð yx
7→ ∇ yx X ( x ) } . Finally, set ∆ w : A −→ A to be given by∆ w f := − div(grad f ) . (2.12)Such ∆ w is called Laplacian operator associated with the w -connection ∇ .Define, for each ( x, y ) ∈ E , the vector field E yx := w ( x, y ) / ∂ yx . (2.13)In what follows, X stands for X ( f ) := X ( X ( f )) for a given X ∈ X and all f ∈ A . Proposition 2.2.
The Laplacian operator ∆ w associated with the w -connectiongiven by (2.10) coincides with the standard weighted Laplacian on a graph. Inother words, ∆ w f ( x ) = X y < x w ( x, y ) (cid:0) f ( y ) − f ( x ) (cid:1) . Proof.
Since { E yx ; y < x } is a g w -orthonormal basis in each tangent space T x V the gradient of a arbitrary function f ∈ A is given bygrad f = X y < x E yx ( f ) E yx Also, by using (2.11) for the w -connection, we obtain (cid:0) ∇ yx E yx ( f ) E yx (cid:1) ( x ) = − E yx ( f ) E yx ( x )= − (cid:0) w ( x, y ) / (cid:1) ∂ yx ( f )( x ) ð yx = − w ( x, y )( f ( y ) − f ( x )) ð yx Thus, ∆ w f ( x ) = − (div(grad f ))( x ) = X y < x w ( x, y ) (cid:0) f ( y ) − f ( x ) (cid:1) . (cid:3) Since Y yx ( f )( x ) = − w ( x, y )( f ( y ) − f ( x )) for all x ∈ V and y < x , the followingcorollary is immediately obtained by Proposition 2.2. Corollary 2.3.
The Laplacian operator ∆ w can be written as ∆ w = − X y < x Y yx . ENERIC SPECTRUM OF THE WEIGHTED LAPLACIAN ON FINITE GROUPS 7
Differential structure of Cayley graphs.
Due to the particular structureof Cayley graphs, they exhibit a regular and well behaved differential structure.Let (
G, . ) be a group finitely generated by S ⊆ G \ { e } . Consider C ( G, S ) to be theassociated Cayley graph as defined in § y < x exactly when thereexists s ∈ S such that y = xs . We can thus write ∂ s ↾ x := ∂ yx for every s ∈ S andall x ∈ G such that y = xs .From this we can easily see that T G ≃ G × T e G with dim( T e G ) = | S | . Let usfix, for each s ∈ S , ∂ s = X x ∈ G ∂ s ↾ x . Thus ∂ s ( f ) = X x ∈ G (cid:0) f ( xs ) − f ( x ) (cid:1) δ x , with ∂ s ( f )( x ) = ∂ s ↾ x ( f )( x ), and ∂ s ( x ) = ∂ s ↾ x ( x ) = ð yx for all x ∈ G and y = xs .Whenever it is convenient, we write the index “ s ↾ x ” instead of “ yx ” when y = xs ( e.g. , Y yx = Y s ↾ x , ð yx = ð s ↾ x ). Define the set L S of constant weights along each s ∈ S , i.e. , w ( x, xs ) = w s for all x ∈ G with w s > G by the maps ℓ y , r y : G → G such that ℓ y ( x ) = yx and r y ( x ) = xy for all x, y ∈ G . Then it is immediate that ℓ y and r y are graph automorphisms. Remark . It is easy to check that every tangent space T x V (and so X x ) arecanonically isomorphic to T e V by dℓ x , the differential of the left translation ℓ x .When it is convenient, by abuse of notation, we shall identify ð s ↾ x (or ∂ s ↾ x ) aselements of T e V . 3. Main results
We will restrict our attention to study the generic irreducibility property of theeigenspaces of a class of Laplacian operators ∆ w on finite Cayley graphs. Our goalis to establish an analogous criterion to the one obtained by Schueth [13]. We willfollow the notation used in [13], adapting it to our purposes if necessary.Unless otherwise stated, we will consider that G is finite, generated by S andthe associated weight w will be such that w ∈ L S .Similarly to (2.13), we can define for each s ∈ S and x ∈ G the following vectorfields E s ↾ x = w / s ∂ s ↾ x , E s := w / s ∂ s . (3.1) Y s ↾ x = − w s ∂ s ↾ x , and Y s := − w s ∂ s . As a consequence of Proposition 2.2 and Corollary 2.3, we have:∆ w f = X x ∈ G ; s ∈ S w ( x, xs ) (cid:0) f ◦ r s − f (cid:1) δ x = − X xs < x E s ↾ x ( f ) . (3.2)A representation of the group G on a vector space W is a group homomorphism ρ : G → GL ( W, K ). We define the pullback ρ ∗ : g → gl ( W ) to be the linearoperator such that ρ ∗ ( ∂ s )( f ) = ∂ s (cid:0) ρ ( e ) f (cid:1) . Therefore, ρ ∗ is given by linear extensionof ρ ∗ ( ∂ s ) = ρ ( s ) − ρ ( e ). C. F. COLETTI, L. R. DE LIMA, D. S. DE OLIVEIRA, AND M. A. M. MARROCOS
The right regular representation R on A is given by R ( y )( f ) = f ◦ r y . Therefore,one can easily see from (3.2) that∆ w f = − X s ∈ S R ∗ ( Y s )( f ) . The left regular representation L on A is given by L ( y )( f ) = f ◦ ℓ y − . Let U asubspace of A . A K -linear operator T : U → U is called
G-simple if its eigenspacesare irreducible subrepresentations of ( A , L ). If K = R we call it real G-simple . Remark . Our goal is to establish conditions in order to determine when ∆ w is areal G -simple operator on the given Cayley graph. Irr ( G, C ) will denote the set of nonequivalent irreducible representations of G . Proposition 3.1.
Let W be a complex irreducible subrepresentation of ( A , R ) .Then ∆ w ( W ) ⊆ W . In particular, for each ( W, ρ ) ∈ Irr ( G, C ) the linear operator ∆ Ww : W → W given by ∆ Ww := − X s ∈ S ρ ∗ ( Y s ) = X s ∈ S w s (cid:0) ρ ( ∂ s ) − id (cid:1) is well-defined.Proof. Consider W subrepresentation of the right regular representation. Then R : G −→ GL ( W ) is well-defined and so is its pullback R ∗ : g −→ gl ( W ). Let f ∈ W . Its Laplacian is given by ∆ w f = − P s ∈ S R ∗ ( Y s )( f ). Since each R ∗ ( Y s )is an operator in gl ( W ), we have that ∆ w f ∈ W . The second assertion of theproposition is a well known fact in representation theory. It ensures that allirreducible representations of G can be seen as a subrepresentation of the rightregular representation up to isomorphism (see for instance [3, Ch. III]). (cid:3) Let (
W, ρ ) ∈ Irr ( G, C ). We know from representation theory of compact groupsthat there exists a G -isomorphism ϕ W : W ∗ ⊗ W → Im ( ϕ W ) ⊆ GL ( A , C ) suchthat, for all x ∈ G , one has id ⊗ ρ ( x ) = ϕ − W ◦ R ( x ) ◦ ϕ W (3.3)(see for instance [3, Ch. III] and [13]). Im ( ϕ W ) is denoted by I ( W ) and it is called the W -isotypical component ofthe right regular representation. Proposition 3.2.
Let W ∈ Irr ( G, C ) . Then, for a given w ∈ L S , id ⊗ ∆ Ww = ϕ − W ◦ ∆ w | I ( W ) ◦ ϕ W . Proof.
Consider h ( x ) = id ⊗ ρ ( x ) and f ( x ) = ϕ − W ◦ R ( x ) ◦ ϕ W . Equation (3.3) tellus that h and f are the same map. By applying the differential of h at e on Y s , wehave dh ( e ) .Y s = id ⊗ ρ ∗ ( Y s ) . Similar arguments apply to the derivative of f at e , which yields df ( e ) .Y s = ϕ − W ◦ R ∗ ( Y s ) ◦ ϕ W . We get to the desired conclusion combining the results above and the sum takenover s ∈ S . (cid:3) ENERIC SPECTRUM OF THE WEIGHTED LAPLACIAN ON FINITE GROUPS 9
Similarly as in [13], we can now describe the Laplacian operator entirely in termsof representation theory over G . Schueth [13] studied the generic irreducibilityproperty using only representation theory of compact groups, which translates forus as finite groups. Thus, we are able to use the pure algebraic results obtained in[13] by means of the propositions 3.1 and 3.2.The next subsection is dedicated to the study of the properties discussed above.The results are analogous to those in [13].3.1. Schueth’s criteria for finite Cayley graphs.
We begin defining types ofcomplex representations.
Definition.
Let (
W, ρ ) be a irreducible complex representation of G . We say that W is of real type (respectively of quaternionic type ) if it admits a conjugatelinear G -map J : W −→ W such that J = id (respectively J = − id ). If W is notof one of those types, it is called of complex type .We will denote by Irr ( G, C ) D the irreducible representations of type D .Let W ∈ Irr ( G, C ). The subrepresentation C W is given by C W := I ( W ) ↔ W ∈ Irr ( G ; C ) R I ( W ) ↔ W ∈ Irr ( G ; C ) H I ( W ) ⊕ I ( W ∗ ) ↔ W ∈ Irr ( G ; C ) C . We also define E W := C W ∩ A R .We state the following result from [13] without proof. In fact, the result does notdepend on the differential properties. Recall that, by (2.12), ∆ w is the divergenceof the gradient associated with the tensor metric g w . One can check that ∆ w hasthe same algebraic properties as the Laplacian with left-invariant metrics in theLie group theory. Furthermore, G can be seen as a Lie group that is a manifold ofdimension 0. Lemma 3.3.
Let W ∈ Irr ( G, C ) and w ∈ L S . Then (i) C W is the complexification of E W . (ii) The following statements are equivalent (A)
Each eigenspace of ∆ w | E W with the action L is in Irr ( G, R ) . (B) Each eigenvalue of ∆ Ww has multiplicity (cid:26) , W ∈ Irr ( G, C ) R , C , W ∈ Irr ( G, C ) H . Corollary 3.4.
Let w ∈ L S . Then ∆ w is real G -simple iff the following statementsare simultaneously satisfied (i) For all W , W ∈ Irr ( G, C ) with W , W ∗ ≇ W , ∆ W w and ∆ W w do notshare any common eigenvalue. (ii) For each W ∈ Irr ( G, C ) R , C , all eigenvalues of ∆ Ww have multiplicity one. (iii) For each W ∈ Irr ( G, C ) H , all eigenvalues of ∆ Ww have multiplicity two. Let g := span R { ∂ s } s ∈ S and suppose { s , ..., s N } an enumeration of S . So g ≃ R N and we represent the coordinates of an arbitrary element in g as α = ( α s , ..., α s N )on the fixed basis { ∂ s j } Nj =1 . We also defines g + = { α ∈ g ; α s j > , j = 1 , ..., N } . Remark . Of course g ≃ R N and g + ≃ L S . Also, g + is a open set in g . Let (
W, ρ ) a representation, α = ( α , ..., α s N ) ∈ g and D W ( α ) : W −→ W givenby D W ( α ) = − X s ∈ S α s (cid:0) ρ ( s ) − id (cid:1) (3.4)Equation (3.4) induces a map D W : g → gl ( W ) for each ( W, ρ ) ∈ Irr ( G, C ). Inparticular, from Proposition 3.1 one has D W (cid:0) w (cid:1) = ∆ Ww . Thus, every left-invariant weighted Laplacian operator can be studied as a D W ( α ) for some α ∈ g + . Let P W : g −→ C [ x ]be the application that maps α ∈ g to the characteristic polynomial of D W ( α ).We can also construct an application res : C [ x ] × C [ x ] −→ C satisfying(i) Given a pair of polynomials p = P j a j x j and q = P k b k x k then res( p, q ) = P j,l,k,m c j,l,k,m a lj b mk for a finite number of coefficients c j,l,k,m .(ii) res( p, q ) = 0 ⇔ p and q do not share any common zero.Such application is called resultant and its existence is a well-known fact (see[8] for more details). Remark . In a few words, item (i) just tell us that the resultant map is a polynomialin the coefficients { a j , b k } j,k of the given pair ( p, q ) of polynomials.Let W, W , W ∈ Irr ( G, C ) and α ∈ g . Denote by P ′ W ( α ) and by P ′′ W ( α ) thefirst and second derivatives of the polynomial P W ( α ), respectively. Consider thefunctions a W ,W , b W , c W : g −→ C given by(i) a W ,W ( α ) = res( P W ( α ) , P W ( α ))(ii) b W ( α ) = res( P W ( α ) , P ′ W ( α ))(ii) c W ( α ) = res( P W ( α ) , P ′′ W ( α )) Remark . Statement (ii) in the definition of resultant says us that a W ,W ( α ) = 0iff D W and D W does not share any common eigenvalue. Similarly, b W ( α ) = 0iff D W has just simple eigenvalues. Finally, c W ( α ) = 0 iff D W has eigenvalues ofmultiplicity at most 2.We can now state the main criterion for the generic simplicity. The result isadapted from Schueth [13] to our theory. Theorem 3.5.
A finite Cayley graph C ( G, S ) admits a weight w ∈ L S such that ∆ w is real G -simple iff the following conditions are simultaneously satisfied (i) a W ,W for all W , W ∈ Irr ( G, C ) with W , W ∗ = W . (ii) b W for all W ∈ Irr ( G, C ) R , C . (iii) c W for all W ∈ Irr ( G, C ) H .Moreover, the existence implies that the set of all the invariant weights w ∈ L S that turns ∆ w into a real G -simple operator constitute a residual set in L S . ENERIC SPECTRUM OF THE WEIGHTED LAPLACIAN ON FINITE GROUPS 11
Proof. If w ∈ L S ≃ g + is such that ∆ w is real G -simple, then ( i ), ( ii ), and ( iii )follow immediately from Corollary 3.4. Now, assume that ( i ), ( ii ), and ( iii ) arejointly satisfied. Note that the elements α ∈ g for that this hypothesis does not holdare exactly those that are in the inverse image at zero of any of those polynomials a W ,W , or b W , or c W stated in items ( i ), ( ii ), and ( iii ). But since Irr ( G, C ) isfinite for the finite group G (see [14, Thm. 7]). Then the inverse image of thesepolynomials at 0 yields a meager finite set N in g . We also have that N ∩ g + ameager set in g + . Thus the set g + \N is residual in g + ≃ L S . Moreover, Corollary3.4 also ensures that g + \N contains exactly the weights α that turns ∆ α into a real G -simple operator. (cid:3) Remark . Note that ∆ w depends on the choice of S ⊆ G \{ e } , and so does a W ,W, , b W and c W . However, once the the generic simplicity holds for S , it follows fromTheorem 3.5 that it also holds for every H ⊆ G \ { e } containing S .4. Applications of Theorem 3.5
Example 1.
Let G be a finite Abelian group. It is known that it is equivalent tothe case where every ( W, ρ ) ∈ Irr ( G, C ) has dimension 1 (see [14], § W j , ρ j ), j ∈ { , , . . . , | G | − } , all distinct irreducible representations of G . Then,one can see that, for all α ∈ g , D W j ( α ) := X s ∈ S α s ( ρ j ( s ) − ρ j ( e )) (4.1)is a scalar. Set κ j to be the constant κ j := D W j ( α ). Therefore P W j ( α )( x ) = x − κ j for all j ∈ { , , . . . , | G | − } . We apply Theorem 3.5. Since the eigenvalues aresimple, , items ( ii ) and ( iii ) are satisfied.Then it suffices to verify if there exists α ∈ g such that κ j = κ k (4.2)for all distinct j, k ∈ { , , . . . , | G | − } . Thus, the generic property for theeigenspaces of ∆ w is conditioned to the existence of α ∈ g satisfying (4.2). Example 2.
Let C n = h a i denote the cyclic group of order n . We know from[14, § n irreducible linear representations are given by ρ j ( a k ) = e π jkn i . Given that S ⊆ G \ { e } is such that there exists ¯ a ∈ S with G = h ¯ a i . Fix ω := ρ (¯ a ), α ¯ a := 1 and α s := 0 for s ∈ S \ { ¯ a } . Since ω j − = ω k − j, k ∈ { , , . . . , n − } , (4.2) holds. Therefore, the subset of L S thatturns ∆ w into a real G -simple operator on C ( C n , S ) is residual for a S described asabove. Example 3.
Let D n the dihedral group of order 2 n . We know from [14, § n is odd and four if n is even) and the others are of degree 2.We will consider S containing any reflection s and any rotation r such that r generates the cyclic group C n = h r i . Thus D n = C n ∪ sC n where sC n denotes the set { sr k ; k = 0 , . . . , n − } . Note that { r, s } ⊆ S is itself agenerator set D n .The irreducible representations of degree 1 satisfy on the generators { r, s } ρ kj ( r ) = k, and ρ kj ( s ) = j where ( k, j ) ∈ { (1 , , (1 , − } if n is oddand ( k, j ) ∈ { (1 , , (1 , − , ( − , , ( − , − } if n is even. We will denote theserepresentations by W kj .The irreducible representations of degree 2 W m = ( C , ρ m ) are given by ρ m ( r k ) = (cid:18) a mk a − mk (cid:19) , and ρ m ( sr k ) = (cid:18) a − mk a mk (cid:19) for 0 < m < n/ a = e πi/n .Let α ∈ g such that α r = 1 and α s = 0 if s = r . Thus D W m ( α ) = (cid:18) a m − a − m − (cid:19) has distinct eigenvalues a m − a − m − a m ′ − a − m ′ − D W m ′ ( α ) for m = m ′ .Thus condition ( i ) of Theorem 3.5 holds for any pair of irreducible representationsof degree 2. In other words a W m ,W ′ m m = m ′ .We must show that the same occurs when we compare the representations ofdegree 1 with representations of degree 2. In fact D W kj ( α ) = ( k − , k = 1 , − . Since 0 < m < n/ πm/n ∈ (0 , π ). Thus we conclude that a m − = k − a W m ,W kj β ∈ g with β r = 2, β s = 1 and β s j = 0 if s j = r, s .Then D W kj ( β ) = 2 k + j − . • ( k, j ) = (1 ,
1) implies that D W kj ( β ) has eigenvalue 0. • ( k, j ) = (1 , −
1) implies that D W kj ( β ) has eigenvalue − • ( k, j ) = ( − ,
1) implies that D W kj ( β ) has eigenvalue − • ( k, j ) = ( − , −
1) implies that D W kj ( β ) has eigenvalue − k, j ) = ( k ′ , j ′ ) one has a W kj ,W k ′ j ′
0. Now condition ( i ) is completelysatisfied.Conditions ( ii ) and ( iii ) of Theorem 3.5 are trivial since all previous operatorshave simple eigenvalues. Example 4.
The alternating group A can be defined by A = h t, x i with t = (123) and x = (12)(34). Let t, x ∈ S . There exists four irreduciblerepresentations (see [14, § ρ j ( t ) = ω j and ρ j ( x ) = 1where ω = e πi/ and j ∈ { , , } . The remaining irreducible representation is ofdegree 3. We will denote it by ρ and it can be generated by ρ ( x ) = − − − and ρ ( t ) = − − − . ENERIC SPECTRUM OF THE WEIGHTED LAPLACIAN ON FINITE GROUPS 13
Let α t = α x := 1 and α s := 0 for s ∈ S \ { t, x } . Thence, P W ( α )( λ ) = λ, P W ( α )( λ ) = λ − ( ω − , P W ( α )( λ ) = λ − ( ω − ,and P W ( α )( λ ) = λ + 7 λ + 15 λ + 8 . From this, one can easily verify items ( i ) ( ii ) and ( iii ) of Theorem 3.5. It thusfollows that the generic irreducibility of the eigenspaces of ∆ w holds on C ( A , S ). Example 5.
Let S be the permutation group of four elements { , , , } .The order of S is 4! = 24 and it is generated by adjacent transpositions (12), (23)and (34). Take S any set of generators that contains the permutations (34) and(123). We know from [14, § τ = (34) and σ = (123). Fix α ∈ g such that α τ = − α σ = 1, and α s = 0 for s = τ, σ . If ( W, ρ ) is a irreducible representation then D W ( α ) = − ρ ( τ ) − id ) + ( ρ ( σ ) − id ) = − ρ ( τ ) + ρ ( σ ) + id Now we introduce the five nonequivalent irreducible representations evaluatedin τ and σ (see [12, 14, 17] for more details). Representations of degree 1 ( W , ρ ) satisfies ρ ( τ ) = ρ ( σ ) = 1, so D W ( α ) has eigenvalue λ = 0.( W , ρ ) satisfies ρ ( τ ) = − ρ ( σ ) = 1, so D W ( α ) has eigenvalue λ = 4. Representation of degree 2 ( W , ρ ) satisfies ρ ( τ ) = (cid:18) − − (cid:19) and ρ ( σ ) = (cid:18) − − (cid:19) . Then D W ( α ) has distinct eigenvalues λ = (1+ √ / λ ′ = (1 −√ / λ − λ − Representations of degree 3 ( W , ρ ) satisfies ρ ( τ ) = − and ρ ( σ ) = − − . Then D W ( α ) has the distinct eigenvalues λ , λ ′ and λ ′′ provided by the characteristicpolynomial − λ + 5 λ − λ − W , ρ ) satisfies ρ ( τ ) = and ρ ( σ ) = − − −
11 0 00 0 1 . Then D W ( α ) has distinct eigenvalues λ , λ ′ and λ ′′ provided by the characteristicpolynomial − λ + λ + 7 λ + 8.One can easily check that all of these ten eigenvalues λ ij , λ ′ ij , and λ ′′ ij are distinct.Thus conditions ( i ), ( ii ) and ( iii ) from Theorem 3.5 are satisfied. Example 6.
Let K ≃ C × C denote the Klein four-group . Recall that K = h a, b i consists of 4 elements such that a = b = ( ab ) = e . We can verify thatthe four linear irreducible representations of K are e a b abρ ρ − − ρ − − ρ − − We consider S to be a generator of G . Fix, without loss of generality, r, t ∈ S to be distinct. We may choose α r = 1. Set α t = 2 α r and α s = 0 for the possibleremaining s ∈ S \ { r, t } . Therefore, the condition (4.2) is verified and the genericproperty holds for every generator set S . eaba b Figure 1.
Tetrahedron associated with the Cayley graph of theKlein four-group generated by the set S = { a, b, ab } .Observe that C ( K , S ) from Example 6 is equivalent to an undirected Cayleygraph. We can use Theorem 3.5 because all t ∈ K is of order 2. We now proceedwith an analogous criterion for more general undirected Cayley graphs.5. Criterion for undirected Cayley graphs
The criterion for the generic simplicity given above can be adapted for undirectedCayley graphs. Recall that C ( G, S ) is equivalent to an undirected graph when S issymmetric, i.e. , when s ∈ S implies that s − ∈ S . The associated weight w ∈ L S must be such that w ( x, xs ) = w ( x, xs − ) for every x ∈ G and s ∈ S . Let us denoteby L ′ S the set of all weights w ∈ L S associated with the undirected Cayley graph C ( G, S ).Set g ′ := { α ∈ g : α s = α s − } and g ′ + := g ′ ∩ g + , then g ′ + ≃ L ′ S . Let W ⊆ g + bethe set of all α ∈ g + such that ∆ α is real G -simple. Similarly, define W ′ := W ∩ g ′ + .Thus Theorem 3.5 leads us to the following theorem for undirected Cayley graphs. Theorem 5.1.
A finite undirected Cayley graph C ( G, S ) admits a weight w ′ ∈ L ′ S such that ∆ w ′ is real G -simple iff the following conditions are simultaneouslysatisfied (i) a W ,W | g ′ for all W , W ∈ Irr ( G, C ) with W , W ∗ = W . (ii) b W | g ′ for all W ∈ Irr ( G, C ) R , C . (iii) c W | g ′ for all W ∈ Irr ( G, C ) H .Moreover, the existence implies that the set of all the invariant weights w ′ ∈ L ′ S that turns ∆ w ′ into a real G -simple operator constitute a residual set in L ′ S .Proof. It suffices to follow the same steps from Theorem 3.5 for g ′ , g ′ + , W ′ and L ′ S instead of g , g + , W and L S , respectively. (cid:3) Remark . Once W ′ is residual in g ′ + , it follows from Theorem 3.5 that W is residualin g + . ENERIC SPECTRUM OF THE WEIGHTED LAPLACIAN ON FINITE GROUPS 15
Note that Remark 6 and the conditions given by Example 1 still hold in theundirected case.
Example 7.
Let C n be the cyclic group of order n >
2. Then for every symmetric S ⊆ C n , C ( C n , S ) does not admit w ′ ∈ L ′ S such that ∆ w ′ is real G -simple.Recall from Example 2 that, if C n = h a i , the irreducible representations can begiven by ρ j ( a k ) = e π jkn i =: ω jk . Let S ⊆ C n \ { e } be symmetric. Since α s = α s − and ( a k ) − = a n − k , it follows that κ = X a k ∈ S α a k ( ω k −
1) = X a k ∈ S α a k ( ω − k −
1) = κ n − for every α ∈ g ′ . Thus, the desired conclusion follows from (4.2) and Theorem 5.1. ea a a ea a a Figure 2.
The complete four graph associated with theundirected (on the left) and the directed (on the right) C ( C , S )where S = { a, a , a } . Remark . The example above shows us a case where we do not have the genericproperty. However, when G is Abelian we can find a weaker version of ourconstruction in such a way that we will reacquire the generic property for theLaplacian.In fact, the condition w ( x, xs ) = w ( x, xs − ) is too strong when considering allfunctions in A . We must then consider a subspace A S of A that is more compatiblewith this property. More specifically, we want the identity w s ( f ( xs ) − f ( x )) = w s − ( f ( xs − ) − f ( x )) . (I)If (I) is true, then the terms associated with w s and w s − will produce the sameeffect in the Laplacian.Map s and s − to the same fixed element s ′ ∈ { s, s − } and repeat this processalong of all the set S . These identifications induces a proper subset S ′ ⊆ S containing all s ′ ∈ S defined as above. An arbitrary weight w = ( w s ) s ∈ S ∈ L ′ S is now completely characterized by the scalars w s ′ with s ′ ∈ S ′ .Define, for each representation ( W, ρ ), the subrepresentation A ρS := \ s ∈ S ker (cid:0) ρ ( s ) − ρ ( s − ) (cid:1) . We note that condition (I) holds exactly for the functions f ∈ A RS with respect tothe right regular representation on A . Take ˆ G the set of nonequivalent irreducible representations in the form ( A ρS , ρ ).Now we can replace ∆ w ′ by ∆ w ′ | A RS and Irr ( G, C ) by ˆ G in Theorem 5.1.Let C n = h ¯ a i and S containing ¯ a . By the previous considerations, Example 7applied to the operator ∆ w ′ | A RS just as ∆ w in Example 2. Therefore, the subset of L ′ S that turns ∆ w ′ | A RS into a real G -simple operator on C ( C n , S ) is residual.6. The operator L w Similarly to the previous section, we can canonically identify the space g K :=span K { ∂ s : s ∈ S } as T e V . For each representation ( W, ρ ), we have ρ ∗ ( ∂ s ) = ρ ( ∂ s ) − id . That induces us to define the linear operators L w and L wW in the samefashion as the Laplacian.Set L w = − X s ∈ S ( R ∗ ( E s )) and L W w = − X s ∈ S ( ρ ∗ ( E s )) . One can easily check that L w f ( x ) = − X s ∈ S w ( x, xs )( f ( xs ) − f ( xs ) + f ( x )) . We shall now study the meaning of this new operator L w and its relation withour theory. First, we say that X ∈ X is a left-invariant vector field if, for each x, y ∈ V , the following equation holds: X ◦ ℓ x ( y ) = dℓ x ( y ) .X ( y ) . (6.1)Here, dℓ x stands for the differential of the graph automorphism ℓ x as in (2.2). Notethat equation (6.1) is equivalent to X ( x ) = dℓ x ( e ) .X ( e ) . Lemma 6.1. X is a left-invariant vector field if, and only if, X ∈ g K .Proof. Suppose X = P s ∈ S X s ∂ s a left-invariant vector field. Then X ( x ) = dℓ x ( e ) .X ( e ) implies that X s ∈ S X s ( x ) ∂ s ( x ) = X s ∈ S X s ( e ) ∂ s ( ℓ x ( e )) = X s ∈ S X s ( e ) ∂ s ( x ) . Hence, each coordinate function X s is constant and then X ∈ g K .Conversely, if X = P s ∈ S a s ∂ s ∈ g for some scalars a s then X ◦ ℓ x ( y ) = X ( xy ) = X s ∈ S a s ∂ s ( xy ) = X s ∈ S a s ∂ s ( ℓ x ( y )) = dℓ x ( y ) .X ( y ) . (cid:3) Corollary 6.2.
The vector fields E s for s ∈ S form a basis for the space of allleft-invariant vector fields. We now compare some results with the Lie groups theory. For instance, thedefinition and properties about left-invariant vector fields coincide. In the Lie
ENERIC SPECTRUM OF THE WEIGHTED LAPLACIAN ON FINITE GROUPS 17 groups theory, we know that if g is a left-invariant metric on a Lie group G ′ then∆ g is given by ∆ g = − m X k =1 E k = − m X k =1 (cid:0) R ∗ ( E k ) (cid:1) where { E k } mk =1 is g -orthonormal basis for its Lie algebra Lie( G ′ ). Here, Lie( G ′ ) isexactly the space of left-invariant fields on G ′ .Back to our context, g K is the analogous to the space Lie( G ′ ) and we can easilycheck from (2.5) and (3.1) that { E s } s ∈ S satisfies g w ( E s , E r ) = δ rs .Of course, the operators ∆ w and L w does not coincide for a given weight w .On the other hand, we saw that the operator L w on Cayley graphs is yet morecomparable to the Laplacian on a Lie group.Another interesting fact is that we can also see L w as the divergence of thegradient in a suitable perspective. We may replace our notion of w -connection fora more akin to the Levi-Civita connection with a left-invariant metric g in the Liegroups case.If { E k } mk =1 is a g -orthonormal basis for Lie( G ′ ), then the Levi-Civita connection ∇ is K -bilinear and satisfies the Leibniz rule ∇ E i ( f E j ) = E i ( f ) E j + f ∇ E i E j . for each f ∈ C ∞ ( G ′ , K ). Moreover, since the Christoffel symbols vanishes in theframe { E k } mk =1 , the equation simply reduces to ∇ E i ( f E j ) = E i ( f ) E j . Then we consider ˜ ∇ to be any K -bilinear map on X that satisfies, for each r, s ∈ S and every f ∈ A K , ˜ ∇ E s ( f E r ) = E s ( f ) E r . Thus, if we take the divergence of the gradient of f by applying this newconnection and the metric tensor g w , we conclude that: L w f = − div(grad f ) . Remark . Of course this identity does not coincide with that presented in (2.12)since the divergence is a concept tied to the given connection.Construct for each (
W, ρ ) ∈ Irr ( G, C ) an operator D W : g → gl ( W ) by D W ( α ) := − X s ∈ S α s (cid:0) ρ ( s ) − ρ ( s ) + ρ ( e ) (cid:1) In particular, D W ( w ) = L wW .Corollary 3.4 holds in exactly the same fashion for the operator L w . We also canreplace D W by D W from (3.4) onward to the end of the main criteria. Thus thefollowing theorem is straightforward: Theorem 6.3.
Theorems 3.5 and 5.1 are also true for the operator L w . We can immediately see that remarks 6 and 7 are also true for L w . Remark . Example 1 for finite Abelian groups is still valid writting D W ( α ) = κ j for κ j := X s ∈ S α j (cid:0) ρ j ( s ) − ρ j ( s ) − ρ ( e ) (cid:1) (6.2)= − X s ∈ S α j (cid:0) − ρ j ( s ) (cid:1) instead of (4.1). Example 8.
Let S ⊆ G \ { e } be sucht that s = e for every s ∈ S . Then G ≃ C × C × · · · × C ( e.g. , K the Klein four group). Therefore L w = 2∆ w , C ( G, S ) is equivalent to an undirected graph an the generic spectrum is completelyconditioned to Theorem 3.5.
Example 9.
Let G = C n be the cyclic group of order n . Recall examples 2 and7. We saw that, for any w ∈ L ′ S and every S ⊆ G \ { e } , ∆ w is not real G -simple.On the other hand, given that there exists ¯ a ∈ S such that C n = h ¯ a i , ∆ w is real G -simple for all w ∈ W and W is residual in L S .Similarly, the same results follow for L w , where κ j = − X ¯ a k ∈ S α ¯ a k (1 − ω kj ) from (6.2) with ω := ρ (¯ a ). References [1]
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Centro de Matem´atica, Computac¸˜ao e Cognic¸˜ao, Universidade Federal do ABC, Av.dos Estados, 5001, 09210-580 Santo Andr´e, S˜ao Paulo, Brazil.
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