Dynamical Decomposition of Bilinear Control Systems subject to Symmetries
DDynamical Decomposition of Bilinear Control Systems subject toSymmetries
Domenico D’Alessandro Jonas T. HartwigDecember 3, 2019
Abstract
We describe a method to analyze and decompose the dynamics of a bilinear control system subject tosymmetries. The method is based on the concept of generalized Young symmetrizers of representation theory.It naturally applies to the situation where the system evolves on a tensor product space and there exists a finitegroup of symmetries for the dynamics which interchanges the various factors. This is the case for quantummechanical multipartite systems, such as spin networks, where each factor of the tensor product represents thestate of one of the component systems. We present several examples of application.
Keywords:
Decomposition of Dynamics; Symmetries; Applications of Representation Theory to Control;Bilinear Systems on Lie groups; Control of Quantum Mechanical Systems.
In geometric control theory, one often considers bilinear systems of the form˙ X = AX + m (cid:88) j =1 B j u j X, X (0) = , (1)where X varies in a matrix Lie group and A and B j ’s belong to the corresponding Lie algebra, with u j the controls,and the identity of the group. It is a well known fact [17] that the reachable set for (1) is the connected Lie group e L , containing the identity , corresponding to the Lie algebra L generated by A and B j ’s, assuming that e L iscompact. Therefore system (1) is called controllable if e L is some ‘natural’ Lie group where the system is supposedto evolve. Common examples are the special orthogonal group SO ( N ) and the unitary group U ( N ) which appearsin applications of control theory to quantum mechanics. If the system of interest has the form˙ ψ = Aψ + m (cid:88) j =1 B j u j ψ, ψ (0) = ψ , (2)where ψ belongs to a vector space ˜ V , the reachable set from ψ is { Xψ | X ∈ e L } . This fact has had manyapplications. In particular, for controlled quantum mechanical systems , in finite dimensions, the equation (1)-(2) isthe Schr¨odinger equation incorporating a semiclassical control field (cid:126)u ( t ) := ( u , ..., u m )) (see, e.g., [7] for examples ofmodeling). In this case, the matrices A and B j in (1), (2) belong to the Lie algebra u ( N ) of skew-Hermitian, N × N ,matrices, so that L is a Lie subalgebra of u ( N ). The matrix X in (1) is called the (quantum mechanical) evolutionoperator and ψ is the state of the quantum system belonging to a Hilbert space ˜ V . In this case, controllability issaid to be verified if e L is the full unitary ( U ( N )) or special unitary ( SU ( N )) Lie group.Although controllability is a generic property (see, e.g., [4], [18]), often, in reality, symmetries of the physicalsystem and a too small number of control functions as compared to the dimension of the system cause the dynamicalLie algebra L , generated by A and B j ’s, to be only a proper Lie subalgebra of the natural Lie algebra associated1 a r X i v : . [ qu a n t - ph ] D ec o the model (for example u ( N )). The problem therefore arises to analyze the structure of this Lie algebra and tounderstand how this impacts the dynamics of the system (1)-(2).In the context of control of quantum systems, which is the main area of application we have in mind, thisproblem has been tackled in several references with tools of Lie algebras and representation theory (see, e.g., [20],[28], [29]). One sees the vector space ˜ V where ψ in (2) lives as the space associated to a representation (see basicdefinitions of representation theory in the next section) of the Lie group e L or the Lie algebra L . In the paper [8],one assumes to have a basis of the dynamical Lie algebra L . Algorithms are given to decompose such a Lie algebrainto Abelian and simple ideals which are its elementary components (Lie sub-algebras). Such algorithms are, forthe most part, simplified and adapted versions of general algorithms presented for Lie algebras over arbitrary fieldsin the book [9]. The paper [20] identifies two causes of uncontrollability for quantum systems. On one hand, thepresence of symmetries, i.e., operators commuting with the full dynamical Lie algebra L , implies that the givenrepresentation of L is not irreducible , that is, the vector space ˜ V , where ψ in (2) lives, splits into a number ofinvariant subspaces each carrying an irreducible representation of the dynamical Lie algebra L . Transitions fromone subspace to the other are forbidden for the dynamics which results in uncontrollability. The second cause ofuncontrollability is the fact that, even within the invariant subspaces, the system might be not controllable becauseof lack of control power. In fact, the paper [20] presents a list of possible Lie subalgebra that might appear asirreducible restrictions of L to invariant subspaces. In view of the recalled decomposition of the dynamical Liealgebra into irreducible components, a new, weaker, notion of controllability was introduced for quantum systemscalled subspace controllability . This is verified when the dynamical Lie algebra is such as to act as u ( n ) on allor some of the invariant subspaces. Subspace controllability was recently investigated for a number of quantumcontrol systems, most notably networks of spins [6], [25], [26]. It was shown [25] that, in some cases, the dimensionof the largest invariant subspace grows exponentially with the number of particles in the network so, subspacecontrollability gives the opportunity of doing universal quantum computation on a restricted subspace even in theabsence of full controllability.From a practical point of view, for a quantum control system with a group of symmetries G , the question arisesof how to obtain the decomposition of the dynamics into invariant subspaces. This is the topic of this paper. Wefocus on a specific method to obtain this which exploits the duality between representations of L and representationsof G (this is some times referred to as Schur-Weyl duality (cf. e.g., [14])). However, in this introduction, we nextdescribe a general different method and then we discuss the drawbacks of this method to motivate instead thetreatment of the rest of the paper.Given the dynamical Lie algebra L ⊆ u ( N ), one calculates a basis of the commutant of L in u ( N ), i.e., thesubspace of u ( N ) of elements that commute with L . This amounts to the solution of a system of linear equations.Being a subalgebra of u ( N ) the commutant is a reductive Lie algebra (see, e.g., [8]), that is, it is the direct sum(i.e., vector space sum of commuting subspaces) of an Abelian subalgebra and a semisimple one. As such, it admitsa
Cartan subalgebra which is a maximal Abelian subalgebra and can be calculated with, for example, the algorithmsof [8], [9]. Elements of a basis of such a Cartan subalgebra can be simultaneously diagonalized and thereforea basis can be found so that they can be written as diag ( i , , ..., ), diag ( , i , ..., ),..., diag ( , , ..., i ), forappropriate dimensions of the zero matrices and the identity matrices . This basis, gives the sought for changeof coordinates that transforms the Lie algebra L in block diagonal form, so that every block corresponds to anirreducible representation of L . In fact, having to commute with the above matrices, the matrices of L take a blockdiagonal form. Moreover, each block corresponds to an irreducible representation of L . To see this, let N be thedimension of such a block, and assume without loss of generality that it is the first block. If this was not irreducible,there would be another block diagonal matrix A in u ( N ), which, in appropriate coordinates, would have all blocksequal to zero and the first block equal to diag ( − ia d , − ib d ) for a (cid:54) = b and appropriate dimensions d and d ofthe identity blocks. The matrix A would be commuting with all the matrices in the dynamical Lie algebra L andwould be also commuting with all the matrices in the above Cartan subalgebra of the commutant. However thiscontradicts the fact that the Cartan subalgebra is maximal Abelian.The above method always gives a basis such that the dynamical Lie algebra L is decomposed into its irreduciblecomponents. However it requires the explicit solutions of linear systems of equations for matrices of possibly highdimension. For example, in the case of a network of n spin particles, the dimension of the state space increasesas 2 n and therefore the above computations involve matrices in u (2 n ), a space of dimension 4 n . Moreover the roleof the group of symmetries G is hidden when we transform the problem into a (high dimensional) linear algebraproblem. For example, if the system is a network of spin ’s and the symmetry group is some subgroup of thesymmetric group (the permutations which leave the matrices appearing in (1) (2) unchanged) such a symmetry2roup is suggested by the topology of the network.This paper is devoted to presenting an alternative to the above approach based to the study of the representationtheory of the symmetry group G itself. The representation theory of finite groups is a topic for which much isknown (see, e.g., [10], [13], [14], [15], [21], [22], [24], [27]). From the knowledge of the representations of the group ofsymmetries G one obtains the change of coordinates which places the Lie subalgebra of all elements of u ( N ) whichcommute with G , u ( N ) G in a block diagonal form, where each block corresponds to an irreducible representation.Since the dynamical Lie algebra L is a Lie subalgebra of u ( N ) G , it will also be placed in the same block diagonalform.This paper is a survey paper or, perhaps more appropriately, an application paper aimed at presenting knownresults in representation theory in a self-contained fashion so that they can be used by control theorists dealingwith systems of the form (1)-(2), and in particular for quantum systems.The paper is organized as follows: In section 2, we give some background notions from representation theoryincluding the definition and properties of Generalized Young Symmetrizers ( GYS ), which play a crucial role inthe method described. The method for dynamical decomposition is described in section 3. It requires identifyingcertain GYS’s and, in section 4, we discuss how these are obtained in two special cases: the case of the fullsymmetric group S n and the case of Abelian groups. In section 5, we present two examples of applications tospin networks where we use the above techniques to obtain the GYS’s and the decomposition. These results, inparticular extend the results of [3] for fully symmetric spin networks to the case of an arbitrary number n of spins,with the computations for the case n = 4 presented in detail. We shall be interested in representations, ( π, ˜ V ), of groups, G , algebras, A , or Lie algebras, R , on a finite dimen-sional complex inner product space ˜ V of dimensions N which we can identify with C N . The space ˜ V is often calleda G -module (or A -module , or an R -module ). Representations are group, algebra or Lie algebra homomorphismsfrom G , A or R to End ( ˜ V ) the space of endomorphisms on ˜ V , which if ˜ V (cid:39) C N can be identified with the space of N × N matrices with complex entries. Given representations of G , A and R , on the same space ˜ V , we shall denoteby A G or R G the (Lie) subalgebra of elements in A or R , or more precisely of their representation, which commutewith the representation of G . For example, for a quantum control system (1) (2), we are given a representationof the dynamical Lie algebra L generated by the ”Hamiltonians” A and B j ’s which is a subalgebra of u ( N ) anda representation on the same space of a group of symmetries which commute with the elements of L . Therefore L ⊆ u ( N ) G the subalgebra of u ( N ) commuting with G .We fix some notations. We shall denote by End G ( ˜ V ) the space of all endomorphisms of ˜ V commuting with G .Given two representations ( π , ˜ V ) and ( π , ˜ V ), Hom ( ˜ V , ˜ V ) denotes the space of homomorphisms φ : ˜ V → ˜ V , Hom G ( ˜ V , ˜ V ) denotes the subspace of Hom ( ˜ V , ˜ V ) of elements φ ∈ Hom ( ˜ V , ˜ V ) such that φπ ( g ) = π ( g ) φ for every g ∈ G . Such a type of maps is called a G -map . Analogously one can consider A -maps and R -maps, for algebras( A ) and Lie algebras ( R ) representations. If the two representations coincide Hom G ( ˜ V , ˜ V ) coincides with End G ( ˜ V ).Two representations ( π , ˜ V ) and ( π , ˜ V ) are called G -isomorphic if there exists an element in Hom G ( ˜ V , ˜ V ), i.e., a G -map, which is also an isomorphism, a G -isomorphism .Representations of groups are called unitary if their images are unitary matrices. Representations of Lie algebrasare called unitary if their images are skew-Hermitian matrices. A representation ( π, ˜ V ) is called reducible if thereexists a proper nonzero subspace of ˜ V which is invariant under the representation, irreducible if there is no suchsubspace. Representations ( π, ˜ V ), of finite groups as well as those of unitary groups or Lie algebras, are completelyreducible , i.e., they can be decomposed into the direct sum of irreducible representations (see, e.g., [13], [27]).In these cases, ˜ V is the direct sum of invariant subspaces for π , so that the restriction of π to each invariantsubspace is an irreducible representation. In this case, in appropriate coordinates, the matrices π ( x ), for x elementin the group, algebra, or Lie algebra, take a block diagonal form. The finite group case and the case of unitaryrepresentations are the cases that will be of interest for us in this paper.In view of these notions, the problem to be solved in this paper, that we have outlined in the introduction, isas follows: 3 roblem: Given a unitary representation of a Lie algebra R , and a unitary representation of a finite symmetry group G , ona finite dimensional Hilbert space ˜ V , find a decomposition of R G into its irreducible components and the associatedchange of coordinates in ˜ V . In the case of quantum control, the Lie algebra R is u ( N ) and if the dynamical Lie algebra L ⊆ u ( N ) commuteswith a group of symmetries G , then we look for a decomposition in irreducible representations of u ( N ) G since L ⊆ u ( N ) G . In the coordinates we find, L also takes a block diagonal form.A fundamental tool in representation theory is the following Schur’s Lemma (see, e.g., [27], Section 2.1).
Theorem 1. (Schur’s Lemma) Let B , be a group or an algebra or a Lie algebra.1. If a complex representation ( π, ˜ V ) of B is irreducible, all B -maps ˜ V → ˜ V are multiples of the identity map.2. Two irreducible representations ( π , ˜ V ) and ( π , ˜ V ) are such that the space of B -maps is either − dimensionalor -dimensional according to whether the two representations are isomorphic or not.Proof. The two statements are equivalent. If statement 2 holds, than taking ( π , ˜ V ) = ( π , ˜ V ) = ( π, ˜ V ) andnoticing that the identity map is a B -map, we obtain statement 1. Now we prove first statement 1 and then showthat statement 2 follows from it.For any B -map φ between two representations ( π , ˜ V ) and ( π , ˜ V ), the Kernel of φ and the Image of φ areinvariant subspaces for the representations ( π , ˜ V ) and ( π , ˜ V ), respectively. Consider now a B -map, φ for therepresentation ( π, ˜ V ) and let α be an eigenvalue of φ . Then, if is the identity map, ˆ φ α := φ − α is a B -map aswell, and the Kernel of ˆ φ α is not zero. Since ( π, ˜ V ) is irreducible, the Kernel must be all of ˜ V , that is φ − α = 0,which proves the first statement.As for the second statement, assume φ : ˜ V → ˜ V is a B -map. Then because of irreducibility Ker ( φ ) = 0 or Ker ( φ ) = ˜ V and Im ( φ ) = 0 or Im ( φ ) = ˜ V . If Ker ( φ ) = 0 and Im ( φ ) = ˜ V then φ is an isomorphism. In all othercases it is zero. If φ and γ are two isomorphisms, from φπ = π φ and γπ = π γ , we obtain φγ − π = π φγ − which using the first statement implies that φ is a multiple of γ , which proves the second statement.We remark that Schur’s Lemma applies to both real and complex (Lie) algebras as long as the consideredrepresentations are complex, i.e., ˜ V , (or ˜ V , ) are complex vector spaces. We need, in fact, the underlying field tobe algebraically closed in order to be able to always find an eigenvalue for the B -map of part 1. More general andabstract formulations of Schur’s Lemma exist (see, e.g., [14] and references therein). Given a finite group G , the group algebra C [ G ] := (cid:76) Π ∈ G C Π is the complex vector space with basis given by theelements of G equipped with multiplication given by bilinearly extending the group operation. For example, for G = S , the symmetric group of three elements,(12) · (cid:0) λ · (1) + µ · (13) (cid:1) = λ · (12) · (1) + µ · (12) · (13) = λ · (12) + µ · (132) , for λ, µ ∈ C (here and in the following we use the convention of multiplying permutations from right to left, ascompositions of transformations). If ˜ V is a G -module then it is also C [ G ]-module where C [ G ] acts on ˜ V by linearlyextending the action of the group G . If we take as ˜ V exactly C [ G ], the action of G on ˜ V gives a representationof G called the regular representation . The regular representation is, in general, not irreducible and it contains,as irreducible components, all the irreducible representations of the finite group G . More precisely, the followingfundamental fact holds (cf., e.g., [13]): Theorem 2.
Every irreducible representation of a finite group G on a vector space ˜ V is G -isomorphic to oneirreducible representation contained in the regular representation. Irreducible representations may be contained (up to G − isomorphism) more than once in C [ G ]. Their multiplicityis equal to the dimension of the representation. That is, we have (cf., e.g., [13] § C [ G ] = (cid:77) j ( C j ) ⊕ dim C j , (3)4or the irreducible representations C j ⊆ C [ G ] of G which, in particular, implies that (cid:88) j (dim C j ) = dim C [ G ] = | G | , (4)the number of elements in the group G . Definition 2.1. ( Generalized Young Symmetrizers (
GYS ) ) Given a finite group G , a complete set of Gener-alized Young Symmetrizers is a set of elements { P j } , j = 1 , . . . , m , of the associated group algebra C [ G ] satisfyingthe following properties:1. ( Completeness ) = m (cid:88) j =1 P j ; (5)where is the identity of the group.2. ( Orthogonality ) P j P k = δ j,k P j , ∀ j, k ; (6)where δ j,k is the Kronecker delta.3. ( Primitivity ) For every g ∈ G P j gP j = λ g P j , (7)for every P j with λ g a scalar that depends on g .Generalized Young symmetrizers are called a complete set of primitive orthogonal idempotents in ring theory.Their significance in representation theory is that they generate left ideals in the group algebra C [ G ] which corre-spond to irreducible sub-representations of the regular representation of G . In particular given a set of GYS’s, wecan write C [ G ] as C [ G ] = C [ G ] = C [ G ]( (cid:88) j P j ) = C + C + · · · + C q , (8)where C j := C [ G ] P j , j = 1 , . . . , q , is a left ideal of C [ G ] and, in particular, an invariant subspace of G in C [ G ], i.e.,a sub-representation of the regular representation. Fix j ≥ x ∈ C j ∩ C + C + · · · + C j − . Then thereexist A , A , ..., A j in C [ G ] such that x = A j P j = A P + A P + · · · + A j − P j − . Multiplying on the right by P j and using (6) we obtain x = 0. Therefore the sum in (8) is a direct sum of sub-modules, i.e., C [ G ] = (cid:76) qj =1 C j .According to Theorem III.3 of the Appendix III of [24], condition (7) is necessary and sufficient so that the ideal C j is minimal which means that it does not properly contain any other ideal. This is usually expressed by saying thatthe idempotent P j is primitive and in terms of representations it means that the representation associated with C j is irreducible . Furthermore, according to Theorem III.2 in Appendix III of [24], GYS’s, P j , always exist, so thatthe irreducible sub-modules C j of C [ G ] can always be written as C j = C [ G ] P j .Primitive, orthogonal idempotents are called Young Symmetrizers in the context of the symmetric group S n and therefore we use here the terminology ‘Generalized Young Symmetrizers’ to refer to the case of a general finitegroup. In the case of the symmetric group, Young symmetrizers are obtained from Young tableaux as summarizedin many textbooks such as [13], [14], and [24]. We shall review the main points in subsection 4.Another property of GYS’s which we shall require is of being Hermitian . To define this property define aconjugate-linear map on C [ G ], which is denoted by † and it is defined on elements of G , like g † := g − andextended to C [ G ] by conjugate linearity, that is, (cid:16)(cid:80) j a j g j (cid:17) † = (cid:80) j ¯ a j g † j , for g j ∈ G and a j ∈ C . With thisdefinition we may require that the GYS’s are Hermitian, i.e., P j = P † j , j = 1 , , ..., q. (9)In our context, we have a G -module, ˜ V , which is extended by linearity to be a C [ G ]-module. We shall seeelements in the group algebra C [ G ] as operators on the vector space ˜ V . We can view, in particular any GYS { P j } as an operator on ˜ V . For a := (cid:80) j a j g j , we have π ( a ) = (cid:80) j a j π ( g j ) and π ( a † ) = (cid:80) j ¯ a j π ( g † j ) = (cid:80) j ¯ a j ( π ( g j )) − . If5he representation ( π, G ) is unitary ( π ( g j )) − = ( π ( g j )) † so that π ( a † ) = (cid:80) j ¯ a j ( π ( g j )) † , so that π ( a † ) = ( π ( a )) † .So if a is Hermitian ( a = a † ), its image under a unitary representation will also be Hermitian in the standard senseof Hermitian matrices.The Hermiticity property will be important in our treatment of representations of Lie subalgebras of u ( N ),in applications to quantum mechanical systems. We will take advantage of recent results of [2] and [16] whichshow how to modify the standard procedure to obtain Young Symmetrizers in order to obtain Hermitian
YoungSymmetrizers, for the case of the symmetric group.
We now assume that, for a group G , we have a complete set of GYS’s. We show how this information can be usedto decompose a Lie algebra R G , i.e., the subalgebra of a Lie algebra R consisting of all elements of R commutingwith G . This gives the decomposition of the dynamics induced by the symmetries in G , and in the associatedcoordinates, the system (2) (and (1)) can be put in a block diagonal form. We shall discuss in the following sectionhow GYS’s can, in certain cases, be obtained.When we are given a system (2) with ψ varying in a complex vector space ˜ V , the space ˜ V simultaneously carriesrepresentations of the dynamical Lie algebra L , a natural Lie algebra R (for example u ( N )), with L ⊆ R , a finitegroup of symmetries G , its group algebra C [ G ], as well as End ( ˜ V ) and End G ( ˜ V ). We are ultimately interested in R G ,since L ⊆ R G , but we describe the representation of End G ( ˜ V ) first. Since R G = R ∩
End G ( ˜ V ), the representationof R G is obtained by restricting the elements of the representation of End G ( ˜ V ) to the ones that also belong to therepresentation of R (for example skew-Hermitian matrices if R = u ( N )). Given the complete set of GYS’s, { P j } and their representations (as elements of the group algebra C [ G ]), which, with some abuse of notation, we stilldenote by { P j } , we consider a decomposition of ˜ V as˜ V = ⊕ qj P j ˜ V . (10)To see that this holds, first notice that for ever y ∈ ˜ V , y = ( (cid:80) j P j ) y = (cid:80) j P j y , because of the completeness property(5). Moreover, for j ≥
2, if x ∈ P j ˜ V ∩ ( P ˜ V + P ˜ V + · · · P j − ˜ V ), i.e., x = P j x j = P x + P x + · · · P j − x j − ,applying P j to both sides and using the orthogonality relation (6), we obtain that x = 0, and therefore the sum(10) is a direct sum (cf. (8)). We choose a basis of ˜ V by putting together bases of P ˜ V , P ˜ V ,..., P q ˜ V . Furthermore,we group together bases corresponding to GYS’s, P j , which give isomorphic ideals, C j , in the group algebra C [ G ].We now analyze the matrix representation of elements in End G ( ˜ V ) in this basis. If F ∈ End G ( ˜ V ), then, forevery j , F P j = P j F , and therefore P j ˜ V is invariant under F . This implies that, in the chosen basis, F has a blockdiagonal form F = A . . . A m A B . . . B m B . . . C . . . C m C (11)where we denoted with the same letter blocks corresponding to isomorphic ideals in the group algebra. We remarkthat, depending on the representation at hand, some ideals C j and corresponding GYS’s P j might not be presentin the above decomposition meaning that some P j ˜ V , might be zero.Now, we want to obtain more information on the nature of the submatrices in F in (11) and we want to studythe form of the representation of G in the same basis. From this, the duality between the representations of G and End G ( ˜ V ), will be apparent. This will rely on the following three propositions whose proofs are presented in thenext subsection. 6 roposition 3.1. Two left ideals in C [ G ], C and C , generated by GYS’s P and P , respectively, are G -isomorphicif and only if there exists an r ∈ G such that P rP (cid:54) = 0 . (12) Proposition 3.2.
For each GYS, P j , P j ˜ V is either zero or it is an irreducible representation of End G ( ˜ V ). Proposition 3.3.
Consider two nonzero
End G ( ˜ V )-modules P j ˜ V and P k ˜ V . They are End G ( ˜ V )-isomorphic if andonly if C j and C k are isomorphic as G -modules. In this case, an End G ( ˜ V )-isomorphism, P j ˜ V → P k ˜ V , is given by P k rP j , for any r ∈ G such that P k rP j (cid:54) = 0, which exists because of Proposition 3.1.Before proving these facts, we see how they impact the form of the representation of End G ( ˜ V ) in (11). Sub-blocksof the matrix F corresponding to isomorphic ideals C j , must have the same dimension, according to Proposition3.3. Therefore the blocks A , ..., A m A have all the same dimension in (11), and the same is true for B , ..., B m B ,and so on. Moreover, we can refine our choice of the basis as follows. Let P j ˜ V ,..., P j m ˜ V be a maximal set ofsubspaces in the decomposition isomorphic to each-other. Choose a basis of for P j ˜ V , { x , ..., x d } , and using theisomorphism of Proposition 3.3, choose a basis for P j ˜ V , given by { P j rP j x , ..., P j rP j x d } , for r ∈ G such that P j rP j (cid:54) = 0. If, for b = 1 , ..., d , F x b = (cid:80) d l =1 a lb x l for some coefficients a lb , then F ( P j rP j x b ) = P j rP j F x b = P j rP j d (cid:88) l =1 a lb x l = d (cid:88) l =1 a lb ( P j rP j x l ) . Therefore, the coefficients of the matrix corresponding to F , { a lb } are the same for the actions on P j ˜ V and P j ˜ V ,and the matrix representations are the same. We can repeat this argument for the remaining P j ˜ V ,..., P j m ˜ V , if any,and show that all the matrices A , ..., A m A , in (11) are equal, i.e., A = A = · · · = A m A = A . Repeating the sameargument for all other sets of isomorphic spaces, we find that, in the given basis, the matrices of representations of End G ( ˜ V ) have the form F = m A ⊗ A m B ⊗ B . . . m C ⊗ C (13)where the numbers m A , m B ,..., m C describe how many times isomorphic representations enter the given represen-tation of End G ( ˜ V ). Moreover the matrices A , B ,..., C correspond to irreducible representations of End G ( ˜ V ).We now study the form of the representation of G in the above basis. Fix one GYS, P and let P , ..., P m theGYS’s corresponding to isomorphic End G ( ˜ V )-modules and isomorphic G -submodules in C [ G ] (cf. Proposition 3.3).If g ∈ G and y := P x ∈ P ˜ V , we have gy = gP x = m (cid:88) j =1 P j + (cid:88) j / ∈{ ,...,m } P j gP x = m (cid:88) j =1 P j gP x, where we used the completeness relation (5) and Proposition 3.1. This shows that (cid:76) mj =1 P j ˜ V is invariant under g and therefore (repeating this argument for every set of isomorphic spaces) that the matrix corresponding to g takesa block diagonal form where each block corresponds to a (large) block in (13) and it is of dimension m A d A × m A d A , m B d B × m B d B ,..., m C d C × m C d C . Here the integers m A,B,....,C indicate how many times isomorphic subspaces P j ˜ V appear in the representation of End G ( ˜ V ) (cf. formula (11)), the integers d A,B,...,C denote their dimensions.Let us focus on the first large block and indicate the number of occurrences simply by m and the dimension simplyby d . If (cid:126)e , ..., (cid:126)e d is a basis of P ˜ V , the chosen basis is (cid:126)e , ..., (cid:126)e d , Φ , (cid:126)e , ..., Φ , (cid:126)e d , ...., Φ m, (cid:126)e , ..., Φ m, (cid:126)e d , where Φ j, , j = 2 , ..., m , is the End G ( ˜ V )-isomorphism, P ˜ V → P j ˜ V chosen above. Therefore the basis for (cid:76) mj =1 P j ˜ V , is givenby P j Φ j, (cid:126)e k , j = 1 , ..., m , k = 1 , ..., d , ordered first by j and then by k , where we set Φ , equal to the identitymatrix. Now, for g ∈ G , calculate gP j Φ j, (cid:126)e k . This gives gP j Φ j, (cid:126)e k = m (cid:88) l =1 P l gP j Φ j, (cid:126)e k . P l gP j is either zero or, according to Proposition 3.3, is an End G ( ˜ V )-isomorphism P j ˜ V → P l ˜ V .Therefore, P l gP j Φ j, is an End G ( ˜ V )-isomorphism P ˜ V → P l ˜ V . According to Schur’s Lemma, Theorem 1, the spaceof End G ( ˜ V )-isomorphisms P ˜ V → P l ˜ V is one-dimensional. Therefore P l gP j Φ j, = λ l,j ( g )Φ l, , and we have gP j Φ j, (cid:126)e k = m (cid:88) l =1 λ l,j ( g )Φ l, (cid:126)e k , with λ l,j ( g ) possibly zero for some g ∈ G . Therefore, by defining Λ m = Λ( g ), the m × m matrix { λ l,j ( g ) } , thematrix corresponding to g in the given basis of (cid:76) ml =1 P l ˜ V has the form Λ m ( g ) ⊗ d . Repeating this for every setof isomorphic representations, we find that the representation of g on ˜ V has the form g = Λ Am A ⊗ d A Λ Bm B ⊗ d B . . .Λ Cm C ⊗ d C . (14)Comparing formula (14) with formula (13) reveals the duality of the representations of End G ( ˜ V ) and G . Thecommutativity of the two representations is also made clear in the given basis. Moreover, the dual roles of theintegers m A,B,...,C and d A,B,...,C is also apparent. In the representation of
End G ( ˜ V ), m is the number of isomorphiccopies of a certain P j ˜ V in ˜ V of dimension d . In the representation of G , the roles of m and d are reversed. Thenumber d represents the dimension of a sub-representation of G and m represents how many times it occurs .If the Lie algebra R for which we want to study the representation of R G is not the full End ( ˜ V ), we can take R G = R ∩
End G ( ˜ V ) and take in (13) the matrices A, B, ..., C , so that the full matrices give the representation of R G . For example, if R = u ( N ) and we look for the representation of u ( N ) G , then we will take matrices A, B, ..., C skew-Hermitian but otherwise arbitrary. In this case, it is important to point out that the GYS’s have to givean orthonormal change of coordinates. This is achieved if, in addition to properties (5), (6) and (7), we have theHermitian property (9). The methods to find GYS described in the next section guarantee that this is the case.
For the proof of Proposition 3.1, we follow [24], Theorem III.4 in Appendix III. For the proofs of Propositions3.2 and 3.3, we combine the treatment of [14] (cf. Theorem 4.2.1) which gives the results for
Hom G ( C j , ˜ V ) withTheorem 9.7 of [22] which says that Hom G ( C j , ˜ V ) and P j ˜ V are isomorphic End G ( ˜ V )-modules. Proof.
First assume that (12) holds and consider the G -map Φ( x ) := xP rP , C → C . The fact that this is a G -map follows easily since, ∀ g ∈ G , Φ( gx ) = ( gx ) P rP = g ( xP rP ) = g Φ( x ). We remark that since P rP (cid:54) = 0the map Φ is not zero on C . In fact, Φ( C ) = Φ( C [ G ] P ) = C [ G ] P rP , which in particular contains P rP .Therefore according to Schur lemma, Theorem 1, C and C are G -isomorphic.Viceversa assume that there is a G -isomorphism, Φ : C → C . Then Φ( P ) must be different from zero otherwisewe would have Φ( C ) = Φ( C [ G ] P ) = C [ G ]Φ( P ) = 0. Moreover Φ( P ) = Φ( P ) P = Φ( P P ) P = P Φ( P ) P (cid:54) = 0,where the first equality is due to the fact that Φ( P ) ∈ C and the last one to the fact that Φ is a G -map. Thereforesince there exists an element S in C [ G ] ( S = Φ( P )) such that P SP (cid:54) = 0, there must exist r ∈ G such that P rP (cid:54) = 0. Otherwise we would have P SP = 0 for any S ∈ C [ G ]. Proof.
Assume (cid:126)x ∈ P j ˜ V and (cid:126)y ∈ P j ˜ V both different from zero (we are assuming P j ˜ V (cid:54) = 0). We shall find an element R ∈ End G ( ˜ V ) such that R(cid:126)x = (cid:126)y . Since (cid:126)x and (cid:126)y are arbitrary, this will imply irreducibility of the End G ( ˜ V )-module, P j ˜ V . Consider C j (cid:126)x which is a G -module. The map Φ x : C j → C j (cid:126)x , given by Φ x ( a ) = a(cid:126)x is a G -map. Moreover it isinjective since C j is irreducible (the Kernel would be a sub-representation (cf. Theorem 1)). Since Φ x is surjectiveby definition C j and C j (cid:126)x are G -isomorphic. The same can be said for C j and C j (cid:126)y , with a map Φ y . We have therefore8 G -isomorphism Φ y ◦ Φ − x from C j (cid:126)x to C j (cid:126)y . In particular, Φ − x ( (cid:126)x ) = P j , so that Φ y ◦ Φ − x ( (cid:126)x ) = P j (cid:126)y = (cid:126)y . Let Φ beany linear extension of Φ y ◦ Φ − x to ˜ V . The map R := 1 | G | (cid:88) g ∈ G g Φ g − , (15)is in End G ( ˜ V ) and coincides with Φ on C j (cid:126)x . Applying R to (cid:126)x , we get (cid:126)y . Proof.
First assume that C j and C k are G -isomorphic. Then, according to Proposition 3.1 there exists r ∈ G suchthat P k rP j (cid:54) = 0. The map P k rP j is an End G ( ˜ V )-map and it is not zero on P j ˜ V (otherwise it would be zero on all of˜ V and therefore it would be zero). Because of the irreducibility of P j ˜ V and P k ˜ V , from Schur’s Lemma, it followsthat P j ˜ V and P k ˜ V , are End G ( ˜ V )-isomorphic.Viceversa, assume that P j ˜ V and P k ˜ V , are End G ( ˜ V )-isomorphic, and both non-zero. Let Ψ be an End G ( ˜ V )-isomorphism, Ψ : P j ˜ V → P k ˜ V . Assume by contradiction that C j and C k are not G -isomorphic. We show thatΨ must be necessarily equal to zero, which gives the desired contradiction. Consider (cid:126)x (cid:54) = 0 in P j ˜ V and thecorresponding Ψ( (cid:126)x ) (cid:54) = 0 in P k ˜ V . Consider the (non-zero) spaces C j (cid:126)x and C k Ψ( (cid:126)x ), and consider the G -map between G -modules C j and C j (cid:126)x , Φ x , defined by Φ x ( a ) = a(cid:126)x . Let T := C j (cid:126)x ∩ C k Ψ( (cid:126)x ). The pre-image of T under Φ x is a G -invariant subspace of C j and since C j is an irreducible G -module it must be zero or the whole C j . It cannot bethe whole C j , because that would imply (repeating the same argument for C k ) that C j (cid:126)x = C k Ψ( (cid:126)x ). In particular,it would imply P j (cid:126)x = aP k Ψ( (cid:126)x ) with a ∈ C [ G ]. However, since C j and C k are assumed to be not isomorphic, fromProposition 3.1 we obtain P j (cid:126)x = P j (cid:126)x = P j aP k Ψ( (cid:126)x ) = 0, which is a contradiction. Therefore the subspace T ⊆ ˜ V isthe direct sum, T = C j (cid:126)x ⊕ C k Ψ( (cid:126)x ). Let Π be an element in End ( ˜ V ) which, when restricted to T gives the projectiononto C k Ψ( (cid:126)x ). We can define R ∈ End G ( ˜ V ), by R := | G | (cid:80) g ∈ G g Π g − . The endomorphism R is equal to Π whenrestricted to T . In particular, it is zero on C j (cid:126)x and the identity on C k Ψ( (cid:126)x ). We have0 = Ψ( R(cid:126)x ) = R Ψ( (cid:126)x ) = Ψ( (cid:126)x ) , which gives the desired contradiction. Example 3.4.
Let the group G be the group Q , of unit quaternions {± , ± i, ± j, ± k } with the standard multi-plication between unit quaternions ij = k = − ji , jk = i = − kj , ki = j = − ik . Since it has order 8, in the regularrepresentation there are two isomorphic 2 − dimensional representations, and four non-isomorphic 1 − dimensionalrepresentations. This fact can be inferred using the formula (cid:80) j (dim C j ) = dim C [ G ] = | G | (cf. (3)), along withthe known fact that the number of non-isomorphic representations in the regular representation is equal to thenumber of conjugacy classes in the group (cf., e.g., [23] Theorem 7 in Section 2.5), which is equal to 5 in the case of Q . Denote by χ the 2 − dimensional representation and by χ , χ , χ , χ the four 1 − dimensional representationsin the regular representation. Consider now, for instance, as ˜ V a 7 − dimensional space and assume that the rep-resentation of Q on ˜ V has one 2 − dimensional representation isomorphic to χ , three isomorphic 1 − dimensionalrepresentations isomorphic to χ and two isomorphic 1 − dimensional representations isomorphic to χ . We assumetherefore that, in the coordinates given by the GYS’s, the representation of g ∈ Q is given as g = A × b b b c
00 0 0 0 0 c , (16)for scalar b and c , and 2 × A × . Comparing (16) with formula (14) (and (13)), we see that, in this case, m A = 2 and d A = 1, m B = 1 and d B = 3, and m C = 1 and d C = 2. The matrices that commute with the matrices9n (16) have the form F = a a B ×
00 0 0 C × , (17)for scalar a , and 3 × B × , and 2 × C × . The irreducible representation χ has dimension m A = 2 but it enters one time ( d A = 1) the representation of Q . Dually, there are m A = 2isomorphic representations of End Q ( ˜ V ) which have dimension d A = 1. There are m A = 2 GYS’s corresponding tothe representation χ . They are given by the block diagonal matrices P := (cid:18) (cid:19) , P := . The irreducible representation χ has dimension m B = 1 but enters three times ( d B = 3) the representation of Q .Dually, there is only one ( m B = 1) isomorphic representation of End Q ( ˜ V ) which has dimension d B = 3. There isonly one GYS corresponding to the representation χ , which, in the chosen coordinates, is given by P :=
00 0 . Analogously, the irreducible representation χ has dimension m C = 1 but enters two times ( d C = 2) the repre-sentation of Q . Dually, there is only one ( m C = 1) isomorphic representation of End Q ( ˜ V ) which has dimension d C = 2. There is only one GYS corresponding to the representation χ , which, in the chosen coordinates, is givenby P := (cid:18) (cid:19) . In the above example, we assumed that the representation of the group is already given in the ‘natural’ basisfrom which the expression of the GYS’s was immediately deduced. Our goal was to illustrate the duality betweenthe representation of the group G and the representation of End G ( ˜ V ). In practice, one is given a representation of G , and therefore of C [ G ]. From the knowledge of the GYS’s and from their images under the given representation,one obtains the change of coordinates which transforms the dynamics in the desired form.We now present a simple example of application to a quantum spin network with symmetries. More examplesof applications to this type of setting will be given in section 5. We recall the definition of the Pauli matrices σ x,y,z which will also be used in section 5. σ x := (cid:18) (cid:19) , σ y := (cid:18) i − i (cid:19) , σ z := (cid:18) − (cid:19) . (18) Example 3.5.
In recent years there has been a large interest for the controllability of central spin networks (see,e.g., [5], [29]), i.e., networks of spin particles where one (central) spin of a given type is connected in various waysto spins of a different type, which may represent a bath. The control may be local, on the central spin, or global onall the spins. One possible topology of the network, which we consider here, is a linear chain with the central spinin the middle and connected with two strings of (bath) spins, of the same length. All spins are interacting witheach other via next neighbor interaction which we assume of the Ising type. Figure 1 describes the configurationof such a spin network:Denote by σ jk for k = x, y, z the tensor product of 2 n + 1 identities, with positions numbered from − n to n ,and with only the j -th position occupied by σ k , so that, for example, for n = 2, σ x = ⊗ ⊗ ⊗ σ x ⊗ . TheHamiltonians describing the dynamics of such a system, i.e., A and B j ’s in (1), are iA = n − (cid:88) j =0 σ jz σ j +1 z + − n +1 (cid:88) j =0 σ jz σ j − z , iB x,y,z = iσ x,y,z , (19)with controls u x,y,z representing local x, y, z -components of electromagnetic fields acting on the central spin only.10igure 1: Example of a symmetric spin network with a central spinFor every n , such a system presents a reflection symmetry ˆ R since the transformation j ↔ − j does not modifythe Hamiltonians in (19). Together with the identity, , ˆ R forms a group of symmetries for the system (1),(19).The two operators P S := ( + ˆ R ), P A := ( − ˆ R ) form a complete sets of GYS’s for this group of symmetries. P S ˜ V ( P A ˜ V ) gives all the states which are symmetric (antisymmetric) with respect to the group { , ˆ R } . In thisbasis the Hamiltonians in (19) are written in block diagonal form. The above method assumes that we are able to obtain, for a given group of symmetries G , the corresponding(Hermitian) GYS’s in the associated group algebra C [ G ], without knowing the irreducible modules of C [ G ] inadvance. To the best of our knowledge, there is no general method to achieve this and it has to be done on acase by case basis. After one finds the GYS’s, their image in the given representation of G applied to ˜ V gives thedesired change of coordinates which puts the dynamics in block diagonal form.We now discuss two cases where it is possible to find the GYS’s. In both cases, we assume that the space˜ V is the tensor product of a number n of identical vector spaces V , i.e., ˜ V = V ⊗ n and G is a subgroup of thesymmetric group S n , which permutes the various factors in V ⊗ . The representation of G is unitary in these cases.The situations we shall treat are when G is the full symmetric group, G = S n , and when G is Abelian . G := S n The construction of the GYS is classical in the case where G = S n (see, e.g., [24]) and we survey here the theory.We shall apply it to a system in quantum control in the following section.Conjugacy classes within S n are determined by the cycle type of a permutation, i.e., the number of cycles ofa certain length. For example for n = 9, the permutation (123)(546)(78)(9) has cycle type: 2 for cycles of length3, 1 for length 2 and 1 for length 1. Cycle types also correspond to partitions of n , i.e., sets of positive integernumbers λ := { λ , ..., λ k } with λ ≥ λ ≥ · · · ≥ λ k ≥
1, and λ + λ + · · · λ k = n . For example, the cycle typeof (123)(546)(78)(9) corresponds to the partition of n = 9, (3 , , ,
1) meaning that the permutations (in the givenconjugacy class) have a cycle of length 3 another cycle of length 3, a cycle of length 2 and a cycle of length 1.Partitions are encoded by
Young diagrams which are diagrams composed of boxes in rows of non-decreasing lengthscorresponding to the numbers in the partitions. For example, the partition of 9, (3 , , ,
1) is encoded in the Youngdiagram . As we have recalled in Example 3.4, it is a known fact in the theory of representations of finite groups thatthe number of non-isomorphic irreducible representations of a finite group G in the regular representation is equalto the number of conjugacy classes in G . Therefore, in the case of the symmetric group, S n , the number ofirreducible representations is equal to the number of Young diagrams. In fact, there is a stronger correspondencebetween Young diagrams and irreducible sub-representations of the regular representation. If λ is a partition of n , a standard Young tableaux of shape λ is obtained from the corresponding Young diagram by distributing the11umbers 1 , , . . . , n over the boxes in such a way that each row and column contains a strictly increasing sequence.For example, T := 1 2 53 6 74 89 (20)is a standard Young tableaux of shape λ := (3 , , , λ is denotedby SYT( λ ). Then there is a correspondence between irreducible sub-representations of the regular representation,corresponding to the partition λ (which are all isomorphic), and elements in SYT( λ ). Each representation is givenby C [ G ] P T where P T is the GYS associated to the tableaux T in SYT( λ ). The GYS P T corresponding to a standardYoung tableaux T in SYT( λ ) is obtained as follows: Let R T be the subgroup of S n consisting of all permutations Πwhich preserve the rows of T . Similarly, let C T be the subgroup of S n of all permutations preserving the columnsof T . For example: T := 1 2 5 73 64 98 R T = S { , , , } × S { , } × S { , } , C T = S { , , , } × S { , , } , where we omitted the singleton symmetric groups such as S { } because they are the trivial group. Here, forinstance, S { , , , } is the subgroup of permutations over the elements { , , , } . The row symmetrizer r T andcolumn anti-symmetrizer c T are elements of C [ S n ] defined as follows: r T = (cid:88) σ ∈ R T σ, c T = (cid:88) σ ∈ C T ( sgn ( σ )) σ (21)The Young symmetrizer associated with T , P (cid:48) T , is defined as P (cid:48) T := r T · c T . Let us consider, for example, n = 3 and the Standard Young Tableaux T = 1 23 . Then R T = S { , } and C T = S { , } and r T = + (12) , c T = − (13) ,P (cid:48) T := r T · c T = ( + (12))( − (13)) = − (13) + (12) − (12)(13) = − (13) + (12) − (132) . Young symmetrizers defined this way satisfy, after being divided by a normalization factor, the completenessproperty (5) and the primitivity property (7). Therefore they give irreducible sub-representations of the regularrepresentation. They satisfy the orthogonality property (6), in general, only for small values of n ( n ≤ u (2 n ) G , every block is also skew-Hermitian. The procedure of [16]has been then modified in [2] to make it significantly more efficient, in particular for large values of n . For ourpurposes however it is enough to use the original recursive algorithm of [16]. We shall call the modified HermitianYoung Symmetrizers of [16] the KS-Young symmetrizers (from the last names of the authors of [16]). Given a YoungTableaux T corresponding to a partition of n , let Pre ( T ) be the Young tableaux obtained from T by removingthe box containing the highest number and therefore corresponding to a partition of n −
1. For example, for thetableau T in (20), Pre ( T ) := 1 2 53 6 74 8 . (22)12he KS-Young symmetrizer P T associated with a tableaux T coincides with the standard Young symmetrizer P (cid:48) T ,if n ≤
2. If n >
2, it is obtained recursively as P T = ( P Pre ( T ) ⊗ ) P (cid:48) T ( P Pre ( T ) ⊗ ) . (23)It is proved in [16] that this definition satisfies the requirements (5),(6), (7) and (9).More information can be obtained from the Young tableau T even without calculating the corresponding KS-Young symmetrizer P T . For instance, the dimension of Im ( P T ) is equal to (cf. Lemma 3 in [16])dim( Im P T ) = (cid:81) rl =1 (cid:81) λ l k =1 ( N − l + k ) Hook ( T ) . (24)Here N = dim( V ), r is the number of rows of the Young tableaux, λ l the number of boxes in the l -th row, Hook ( T )is the Hook length of the Young diagram associated with T . It is calculated by considering, for each box of theYoung diagram, the number of boxes directly to the right + the number of boxes directly below + 1 and thentaking the product of all the numbers obtained. For example the Hook length of the Young tableau in (20) is 2160.It follows from formula (24) that if the number of rows of the tableaux is greater than the dimension N of thevector space V , then dim( Im ( P T )) = 0. Let G be a finite Abelian group. It follows from Schur’s Lemma that every irreducible representation is onedimensional. In the following, we shall use some concepts concerning the character χ of a representation ρ (cf.,e.g., Lecture 2 in [13]). This is a function G → C defined as χ ( g ) = Tr ρ ( g ), for g ∈ G . Various properties ofcharacters of representations can be found in the representation theory texts we have cited. One property that wewill use, and that directly follows from the definition, is that the character of the direct sum of two representationsis the sum of the characters (cf. Proposition 2.1 in [13]). Characters corresponding to irreducible (and thereforeone dimensional) representations are called irreducible characters . There is a one to one correspondence betweenirreducible characters and irreducible representations. Every irreducible character is a group homomorphism χ : G → C × , whose image is contained in the unit circle S , in C × , the complex plane without the origin. Recall thatfrom formula (3) (with dim( C j ) = 1) there are | G | different irreducible representations in the regular representationand therefore | G | different characters. To each such character χ we associate an element P χ of the group algebra C [ G ] as follows: P χ := 1 | G | (cid:88) g ∈ G χ ( g ) g. (25) Proposition 4.1.
The set { P χ } where χ ranges over the set of all possible irreducible characters, in the regularrepresentation, forms a complete set of Hermitian GYS for the group G , i.e., it satisfies properties (5)-(7) and (9). Proof.
Consider the following calculation. P χ P χ (cid:48) = 1 | G | (cid:88) g,g (cid:48) ∈ G χ ( g ) χ (cid:48) ( g (cid:48) ) gg (cid:48) = 1 | G | (cid:88) h ∈ G (cid:0) (cid:88) g,g (cid:48) ∈ Ggg (cid:48) = h χ ( g ) χ (cid:48) ( g (cid:48) ) (cid:1) h = 1 | G | (cid:88) h ∈ G (cid:0) | G | (cid:88) g ∈ G χ ( g ) χ (cid:48) ( g − ) (cid:1) χ (cid:48) ( h ) h = 1 | G | δ χχ (cid:48) (cid:88) h ∈ G χ (cid:48) ( h ) h = δ χχ (cid:48) P χ (cid:48) , Since any element of the representation acts as a multiple of the identity, irreducibility can only occur in dimension 1. | G | (cid:80) g ∈ G χ ( g ) χ (cid:48) ( g − ) = δ χχ (cid:48) (cf. formula (2.10) in [13]), withthe property ¯ χ ( g ) = χ ( g − ). This gives (6).To see that P χ is Hermitian, we calculate P † χ = (cid:88) g ∈ G χ ( g ) g − = (cid:88) g ∈ G χ ( g − ) g − = (cid:88) h ∈ G χ ( h ) h = P χ . In the last equality, we used the substitution h = g − .Next, we have (cid:88) χ P χ = 1 | G | (cid:88) χ (cid:88) g ∈ G χ ( g ) g = 1 | G | (cid:88) g ∈ G (cid:0) (cid:88) χ χ ( g ) (cid:1) g (26)The function (cid:80) χ χ , as a function of g , is the character of the regular representation (being the sum of all itsirreducible characters). The matrix associated with g (as a linear transformation on C [ G ]) is a permutation matrixwhich transforms the basis { h | h ∈ G } to { gh | h ∈ G } . Such a permutation has trace zero for any g ∈ G , exceptwhen g is the identity. In that case (cid:80) χ χ ( g ) = | G | , and the right hand side of (26) is equal to the identity.Lastly, we need to show that P χ gP χ = λ g P χ , for some λ g depending on g , i.e., property (7). In fact, we have,since the group G is Abelian, P χ gP χ = P χ P χ g = P χ g = 1 | G | (cid:88) h ∈ G χ ( h ) hg = 1 | G | (cid:88) m ∈ G χ ( mg − ) m =1 | G | (cid:88) m ∈ G χ ( m ) χ ( g − ) m = χ ( g − ) (cid:32) | G | (cid:88) m ∈ G χ ( m ) m (cid:33) = χ ( g − ) P χ , as desired. We now apply the above described method to the analysis of the dynamics of two examples concerning networksof spins.
Consider a network of n identical spin particles under the control action of a common magnetic field and exhibitingidentical Ising interaction with each other [6]. We denote by 0 (cid:105) and | (cid:105) the states of the spin particle, i.e.,the two possible eigenstates when measuring the spin in a given direction (e.g., the z -direction). Since every spininteracts with every other spin in the same way, we call such networks completely symmetric . The state space is V ⊗ n where V = C with the standard inner product (cid:104) φ | ψ (cid:105) := φ ∗ ψ . Schr¨odinger equation for the dynamics is givenby (2) with A = − iH zz and (cid:80) B j u j := − iH x u x − iH y u y , where the quantum mechanical Hamiltonians , H zz , H x and H y , acting on V ⊗ n , are given by H x = (cid:88) ⊗ · · · ⊗ ⊗ σ x ⊗ ⊗ · · · ⊗ , (27) H y = (cid:88) ⊗ · · · ⊗ ⊗ σ y ⊗ ⊗ · · · ⊗ , (28) H zz = (cid:88) ⊗ · · · ⊗ ⊗ σ z ⊗ ⊗ · · · ⊗ ⊗ σ z ⊗ ⊗ · · · ⊗ , (29) u x and u y represent x and y components of the external (semi-classical) control magnetic field and σ x,y,z are thePauli matrices defined in (18). In (27), (28) the sum is taken over all the spins, which are assumed identical, whilein (29) it is taken over all the (cid:0) n (cid:1) pairs of spins. The group of all permutations on n objects, i.e., the symmetricgroup S n , acts as a group of symmetries for this system by permuting the factors in the tensor products:Π( v ⊗ · · · ⊗ v n ) = v Π(1) ⊗ · · · ⊗ v Π( n ) , ∀ Π ∈ S n . (30) See also [11] and [12] for interesting quantum states possibly generated by these systems. u S n (2 n ) := (cid:0) u (2 n ) (cid:1) S n . The three Hamiltonians (27), (28), (29) commute with the action of the symmetricgroup S n . Therefore the dynamical Lie algebra L is a subalgebra of u S n (2 n ). The dimension of u S n (2 n ) wascalculated in [3] to be (cid:0) n +3 n (cid:1) . In fact, it was shown in [3], that the dynamical Lie algebra L in this case is exactlyequal to u S n (2 n ) ∩ su (2 n ), i.e., u S n (2 n ) with the restriction that the trace is equal to 0.Models of this type often represent crystals of identical equidistant particles. The fact that the particles havethe same distance from each other implies that they have the same interaction with each other.The GYS’s and the associated change of coordinates can be calculated with the method of Young tableauxdescribed in subsection 4.1. Here we calculate the explicit change of coordinates for the case n = 4. This case isnot only the simplest case that was not treated in [3] but also the highest dimension physically relevant when weconsider spin networks, since symmetry often requires that the spins are equidistant. Therefore in 3 − dimensionalspace there are at most 4 of them. In the following we denote by S a ,a ,...,a r the symmetrizer of positions a , a , ..., a r and by A a ,a ,...,a r the anti-symmetrizer of positions a , a , ..., a r , i.e., (cf. (21)) S a ,a ,...,a r := (cid:88) σ ∈ S { a ,a ,...,ar } σ, A a ,a ,...,a r := (cid:88) σ ∈ S { a ,a ,...,ar } sgn ( σ ) σ, (31)where S { a ,a ,...,a r } is the permutation group of the symbols { a , a , ..., a r } . We also denote by V j , j = 0 , , , , V ⊗ spanned by states with j , 1’s, so that, for instance, V = span {| (cid:105)} . (4)There is only one Standard Young Tableaux (SYT) corresponding to such a partition given by1 2 3 4 . The corresponding KS-Young Symmetrizer P coincides with the standard Young symmetrizer P (cid:48) (thiscan be shown by induction to be true for every KS-symmetrizer corresponding to partition ( n ) for every n ). Theimage of P is spanned by the symmetric orthogonal states (for simplicity we omit the normalization factor). ϕ = | (cid:105) , (32) ϕ = | (cid:105) + | (cid:105) + | (cid:105) + | (cid:105) , (33) ϕ = | (cid:105) + | (cid:105) + | (cid:105) + | (cid:105) + | (cid:105) + | (cid:105) , (34) ϕ = | (cid:105) + | (cid:105) + | (cid:105) + | (cid:105) , (35) ϕ = | (cid:105) . (36) (3 , , , , . Using the recursive method of [16] described in subsection 4.1 we compute the KS-Young symmetrizers and thecorresponding bases. • For P , we get, up to a multiplicative constant, P = P P (cid:48) P = P (cid:48) P (cid:48) P (cid:48) = S , , A , S , , , which applied to V and V gives zero, while applied to V , , gives the span of ψ , , with ψ = | (cid:105) + | (cid:105) + | (cid:105) − | (cid:105) ψ = | (cid:105) + | (cid:105) + | (cid:105) − | (cid:105) − | (cid:105) − | (cid:105) ψ = | (cid:105) + | (cid:105) + | (cid:105) − | (cid:105) . Notice that ψ is obtained from ψ by exchanging the 1’s with the 0’s.15 For P , we get, up to a multiplicative constant, P = P P (cid:48) P = P P (cid:48) P P (cid:48) P P (cid:48) P = S , A , S , S , , A , S , A , S , = S , A , S , , A , S , A , S , which applied to V and V gives zero, while applied to V , , gives the span of χ , , with χ = | (cid:105) + | (cid:105) − | (cid:105) χ = 2 | (cid:105) − | (cid:105) + | (cid:105) + | (cid:105) − | (cid:105) − | (cid:105) χ = | (cid:105) + | (cid:105) − | (cid:105) . • For P , we get, up to a multiplicative constant, P = P P (cid:48) P = P (cid:48) P (cid:48) P (cid:48) P (cid:48) P (cid:48) P (cid:48) P (cid:48) = A , S , A , A , S , , A , A , S , A , A , = A , S , A , S , , A , S , A , , which applied to V and V gives zero, while applied to V , , gives the span of η , , with η = | (cid:105) − | (cid:105) ,η = | (cid:105) + | (cid:105) − | (cid:105) − | (cid:105) .η = | (cid:105) − | (cid:105) . (2 , , , . Using the algorithm in [16], we compute the KS-Young symmetrizers and the corresponding bases. • For P , we get, up to a multiplicative constant, P = P P (cid:48) P = P (cid:48) P P (cid:48) P (cid:48) P (cid:48) P P (cid:48) = S , A , S , S , A , A , S , A , S , . which applied to V , , , gives zero, while applied to V gives the span of µ = 2 | (cid:105) + 2 | (cid:105) − | (cid:105) − | (cid:105) − | (cid:105) − | (cid:105)• For P , we get, up to a multiplicative constant, P = P P (cid:48) P = P (cid:48) P (cid:48) P (cid:48) P (cid:48) P (cid:48) P (cid:48) P (cid:48) = A , S , A , S , S , A , A , A , S , A , = A , S , A , S , S , A , A , S , A , , which applied to V , , , gives zero, while applied to V gives the span of ν = | (cid:105) + | (cid:105) − | (cid:105) − | (cid:105) . .1.4 Structure of the dynamical Lie algebra L According to the theory developed in this paper, the above change of coordinates transforms the matrices in u S (2 ) into a block diagonal form with one copy of u (5) acting on span { ϕ , ϕ , ϕ , ..., ϕ } , the so called symmetricstates , three copies of u (3) acting respectively on span { ψ , ψ , ψ } , span { χ , χ , χ } , or span { η , η , η } and twocopies of u (1) acting, respectively, on span { µ } or span { ν } . Therefore, in the given coordinates, matrices in L = u S n (2 ) ∩ su S n (2 ) (recall that from the results of [3] the dynamical Lie algebra L is equal to u S n (2 n ) exceptfor the requirement that the matrices have zero trace) have the form (cf. (13)) A × B × B × B × C ×
00 0 0 0 0 C × where A × is an arbitrary matrix in u (5), B × is an arbitrary matrix in u (3) and C × is an arbitrary number in u (1) (i.e., a purely imaginary number), with T r ( A × ) + 3 T r ( B × ) + 2 C × = 0. The system is state controllableon each of the invariant subspaces, that is, it is subspace controllable. We may calculate the matrices of therestrictions of − iH zz , − iH x and − iH y to the various invariant subspaces and consider control theoretic problemsin each subspace. Consider a circular network of identical spin particles interacting via Ising z - z interaction but with nearestneighbor interaction only. The Hamiltonians modeling the interaction with the external magnetic (control) field inthe x and y direction are again given by (27) and (28). However the Hamiltonian modeling the interaction betweenthe particles, H zz in (29), has to be replaced by H NNzz = σ z ⊗ σ z ⊗ ⊗ · · · ⊗ + ⊗ σ z ⊗ σ z ⊗ ⊗ · · · ⊗ + · · · + ⊗ ⊗ · · · ⊗ ⊗ σ z ⊗ σ z + σ z ⊗ ⊗ · · · ⊗ ⊗ σ z . (37)The relevant group of symmetries here is the Abelian subgroup C n of S n , defined as the group generated by thecircular shift { , , ..., n } → { n, , , ..., n − } , i.e., the permutation Z := (123 · · · n ), with Z n = . The dynamicalLie algebra L is a subalgebra of u C n (2 n ) := ( u (2 n )) C n . The dimension of u C n (2 n ) is derived in Appendix A and itis given by dim u C n (2 n ) = 1 n (cid:88) m | n nm φ ( m ) , (38)where (cid:80) m | n means we sum over all positive integers m which divide n , and φ ( m ) is the Euler’s totient function (see, e.g., [1]) defined as φ (1) = 1 and φ ( m ) equal to the number of positive integers less than m which are relativelyprime to m , if m >
1. It is interesting to note that, contrary to what happens in the example of the previoussubsection, the dynamical Lie algebra L in this case may be a proper Lie subalgebra of u C n (2 n ) (modulo therequirement of zero trace). Consider, for instance, the case n = 3. From formula (38) since φ (1) = 1 and φ (3) = 2,we have dim u C (2 ) = 13 (cid:0) × × (cid:1) = 24 . Therefore u C n (2 n ) is larger than u S n (2 n ), since the latter has dimension (cid:0) n +3 n (cid:1) = 20. On the other hand, for n = 3, the dynamical Lie algebra generated by iH NNzz in (37) and iH x and iH y in (27), (28) is the same as the onegenerated by (27), (28) and (29) since the Hamiltonian H NNzz in (37) coincides with the Hamiltonian H zz in (29)in this case. So the dynamical Lie algebra is L = u S n (2 n ) ∩ su (2 n ) in this case because of the result of [3]. Thishas dimension 19 while u C n (2 n ) ∩ su (2 n ) has dimension 23.Since C n is an Abelian group, every finite-dimensional irreducible representation is 1-dimensional. There areexactly n not equivalent such representations (in the regular representation) which we denote by: ρ , ρ , . . . , ρ n − .They are given by ρ k : C n → GL (1 , C ) = C × (39) ρ k ( Z j ) = ε kj (40)17here ε := ε n := e πi/n is the n -th root of the identity. The character associated to the representation ρ k , k = 0 , , , ..., n − χ k ( Z j ) := Tr ρ k ( Z j ) = ε kj . Using Proposition 4.1, a complete set of Hermitian GYS’s is thengiven by the following n Fourier sums in the group algebra C [ C n ]: P k = 1 n n − (cid:88) j =0 χ k ( Z j ) Z j = 1 n n − (cid:88) j =0 ε kj Z j , k = 0 , , . . . , n − . (41) We now want to decompose the Lie algebra u C n (2 n ), which has dimension given in formula (38), using the GYS’s(41). From this, we deduce the decomposition of the dynamical Lie algebra L for the system of n interacting spinwith circular symmetry.Let V = C modeling the state of spin systems. States in a basis of V ⊗ n are labeled by binary words a = a a . . . a n ∈ { , } n as follows: | a (cid:105) = (cid:126)a ⊗ (cid:126)a ⊗ · · · ⊗ (cid:126)a n , (42)where (cid:126) ), (cid:126) ). According to the method of this paper, we need to describe Im ( P k ), for a complete set ofGYS’s { P k } . We notice that the space V ⊗ nT defined as the span of states | a (cid:105) with a a word of period T (necessarily)dividing n , is invariant under C n and therefore it is invariant under the action of any element of the group algebra C [ C n ] including the GYS’s { P k } . The period T is the smallest positive integer such that Z T ( a ) = a . We have( P k V ⊗ n ) = P k (cid:77) T | n V ⊗ nT = (cid:77) T | n ( P k V ⊗ nT ) . (43)Consider a general vector | a (cid:105) in the standard basis of V ⊗ n and belonging to V ⊗ nT . With a GYS, P k , defined in(41), we have P k (cid:0) | a (cid:105) (cid:1) = 1 n n − (cid:88) j =0 ε kj · | a j a j · · · a n + j (cid:105) (44)where the indices of a l are considered modulo n . Since the word a a · · · a n is periodic of period T , that is, a T a T · · · a n + T = a a · · · a n . or Z T ( a ) = a , in the right hand side of (44), we can divide the summationvariable j by T to get j = T q + r, ≤ r < T, ≤ q < nT . (45)Thus P k (cid:0) | a (cid:105) (cid:1) = 1 n T − (cid:88) r =0 (cid:0) nT − (cid:88) q =0 ε kT q + kr (cid:1) · | a r a r · · · a n + r (cid:105) = 1 n T − (cid:88) r =0 ε kr (cid:0) nT − (cid:88) q =0 ε kT q (cid:1) · | a r a r · · · a n + r (cid:105) . (46)The quantity in parenthesis can be computed as a geometric series to give nT − (cid:88) q =0 ε kT q = (cid:40) nT , if ε kT = 1 , ( ε kT ) n/T − ε kT − = 0 , otherwise, (47)because ε n = 1, since by definition ε := e i πn . Using this, we get P k (cid:0) | a (cid:105) (cid:1) = (cid:40) T (cid:80) T − r =0 ε kr · | a r a r · · · a n + r (cid:105) , if ε kT = 1 , , otherwise . (48)Then P k (cid:0) | a (cid:105) (cid:1) is non-zero if and only if ε kT = 1, which happens if and only n/T divides k . Therefore P k V ⊗ nT in(43) is nonzero only if n/T divides k . For a representation ρ , we can write ρ ( Z ) in Jordan canonical form, in appropriate coordinates. From ρ ( Z ) n = ρ ( ) n = it followsthat each Jordan block must be a multiple of the identity, λ with λ an n -th root of the identity. xample 5.1. Consider n = 4, so that V ⊗ n is 16-dimensional. In general the possible values of period (dividing n = 4) are T = 1, T = 2, and T = 4. Let us calculate Im ( P ). All T = 1, T = 2, and T = 4 are such that n/T = 4 /T ,divide k = 0. We have one state for each orbit of C , which gives 6 states (cid:80) j =0 Z j | (cid:105) , (cid:80) j =0 Z j | (cid:105) , (cid:80) j =0 Z j | (cid:105) , (cid:80) j =0 Z j | (cid:105) , (cid:80) j =0 Z j | (cid:105) , and (cid:80) j =0 Z j | (cid:105) , which span Im P . For k = 1 theonly possibility is T = 4, so that n/T = 1. We have the three states: (cid:80) j =0 (cid:15) j Z j | (cid:105) , (cid:80) j =0 (cid:15) j Z j | (cid:105) , (cid:80) j =0 (cid:15) j Z j | (cid:105) . For k = 3 the only possibility is also T = 4, and we also have three states: (cid:80) j =0 (cid:15) j Z j | (cid:105) , (cid:80) j =0 (cid:15) j Z j | (cid:105) , and (cid:80) j =0 (cid:15) j Z j | (cid:105) . For k = 2 the possibilities are T = 4 and T = 2. For T = 4, wealso have three states: (cid:80) j =0 (cid:15) j Z j | (cid:105) , (cid:80) j =0 (cid:15) j Z j | (cid:105) , and (cid:80) j =0 (cid:15) j Z j | (cid:105) . For T = 2, we have onestate (cid:80) j =0 (cid:15) j Z j | (cid:105) . Therefore we have dim( Im P ) = 6, dim( Im P ) = 3, dim( Im P ) = 4, dim( Im P ) = 3,so that u C n (2 ) = u (6) ⊕ u (3) ⊕ u (4) ⊕ u (3), since all the irreducible representations associated to the GYS, P k ,are inequivalent. The dimension of u C n (2 ) which is equal to 6 + 3 + 4 + 3 = 70 can also be calculated usingformula (38), which gives (4 + 4 + 2 × ) = 70.Generalizing the previous example, we now want to calculate the dimension of Im ( P j ), which we denote by m j := dim( Im ( P j )), so that, u C n (2 n ) = u ( m ) ⊕ u ( m ) ⊕ · · · ⊕ u ( m n − ) . (49)Consider the set X k of binary words a of length n and with a period T such that n/T divides k . Since the cyclicgroup C n preserves the period, X k is invariant under C n . The cyclic group C n acts on X k by cyclic permutationsof the letters. Moreover, as we have seen above, P k is non zero only on the vector subspace of V ⊗ n spanned by thevectors corresponding to the words in X k . Similarly to what done in Proposition 5.2 in the Appendix A, there is aone to one correspondence between the orbits of C n in X k and elements in a basis of Im ( P k ) given, using (48), by[( a a · · · a n )] ∈ X k /C n ↔ T T − (cid:88) r =0 ε kr · | a r a r · · · a n + r (cid:105) , (50)which is independent of the representative chosen for [( a a · a n )]. In particular m k = dim Im ( P k ) = | X k /C n | .Using this, we obtain in Appendix A m k = 1 n (cid:88) m | gcd( n,k ) w ( n, k, m ) · φ ( m ) . (51)Here w ( n, k, m ) is the number of binary words a of length n which have a period T such that m divides n/T and n/T divides k . Consider as an example w (6 , , n = 6, possible values for the periods T are T = 1 , , , T = 1, nT = 6, which does not divide k = 4. For T = 2, nT = 3 but m = 2 does not divide nT = 3. For T = 6, nT = 1, but m = 2 does not divide nT = 1. However for T = 3, we have nT = 2. m = 2 divides nT = 2 and nT = 2 divides k = 4. We count the number of binary words of period 3 with 6 elements which are 6. Therefore w (6 , ,
2) = 6. In formula (51) again, as in formula (38), φ ( m ) denotes the Euler’s totient function computed at m .The following is a case where we are able to calculate the dynamical decomposition of u C n (2 n ) explicitly. n is a prime number Suppose n = p where p is a prime number. If k = 0 there are two terms in the sum (51), the one correspondingto m = 1 and the one corresponding to m = p . For m = 1 we can take words of period T = 1 and T = p whichrepresent all possible 2 p words. So we have a term 2 p φ (1) = 2 p in the sum. For m = p we can only take words ofperiod T = 1, since words of period T = p are such that n/T = 1 and m = p does not divide 1. There are only 2such words (000 · · ·
0) and (111 · · · φ ( p ) = 2( p −
1) in the sum. Therefore, we have m = 1 p (2 p + 2( p − p − p . Notice that, for any integer a and prime number p , the quantity a p − a is divisible by p , by Fermat’s Little Theorem(see, e.g., [19]). If k > k , the only possible period in the sum (51) is T = p m is m = 1. Thus there is only one term in the sum corresponding to all wordsexcept the two of period T = 1. We obtain m k = m − p (2 p − , ≤ k < p. (52)Consequently, u C p (2 n ) = u (cid:0) p − /p ) (cid:1) u (cid:0) (2 p − /p (cid:1) . . . u (cid:0) (2 p − /p (cid:1) (53). The dimension is equal todim u C p (2 n ) = (2 p + 2 p − + ( p − p − p = 4 + (4 p − /p (54)after simplification. This also agrees with the formula (38) for n = p , a prime number. As we have discussed above, the dynamical Lie algebra L associated with a circularly symmetric network of spin particles may, in general, be a proper subalgebra of u C n (2 n ). Nevertheless the change of coordinates we haveobtained in this section places L in a block diagonal form from which its structure is easier to understand. Weillustrate this for the case n = 3.Since n = 3 is a prime number, we can use the simplified formula (54) for m = dim( Im ( P )), m = dim( Im ( P )), m = dim( Im ( P )), and we get m = 4, m = 2, m = 2. From (48) we obtain a formula for an orthogonal basisof Im ( P ) which, after normalization, is given by ϕ := | (cid:105) ; ϕ := | (cid:105) ; ϕ := 1 √ | (cid:105) + | (cid:105) + | (cid:105) ) ϕ := 1 √ | (cid:105) + | (cid:105) + | (cid:105) ) . (55)We also obtain a formula for an orthonormal basis of Im ( P ) ( (cid:15) := e i π ) ψ := 1 √ | (cid:105) + (cid:15) | (cid:105) + (cid:15) | (cid:105) ) ψ := 1 √ | (cid:105) + (cid:15) | (cid:105) + (cid:15) | (cid:105) ) , (56)and a formula for an orthonormal basis of Im ( P ), η := 1 √ | (cid:105) + (cid:15) | (cid:105) + (cid:15) | (cid:105) ) η := 1 √ | (cid:105) + (cid:15) | (cid:105) + (cid:15) | (cid:105) ) . (57)By calculating the action of − iH NNzz , − iH x and − iH y in (27), (28), (37) on the above basis, using the fact that 1+ (cid:15) + (cid:15) = 0, we obtain, the expression of these operators in the new basis. This is, − i ˆ H NNzz = diag ( − i, − i, i, i, i, i, i, i ),20nd − i ˆ H x := − i √ − i √ − i √ − i −√ i − i ii ii , − i ˆ H y := √ −√ −√ √ − −
11 0 00 0 0 −
11 0 . The upper left blocks generates any possible 4 × × u C (2 ) (cf. equation (53)). Therefore the dimensionof the dynamical Lie algebra is 4 + 2 − −
1, where the − S as the symmetry group of the model. From this decomposition we can infer further propertiesconcerning the subspace controllability of the system under consideration. We know that the subsystems identifiedby the vectors { ϕ , ϕ , ϕ , ϕ } , { ψ , ψ } , { η , η } , are all state controllable. Therefore we have controllability forany invariant subspace, i.e., subspace controllability. Acknowledgement
Domenico D’Alessandro is supported by the NSF under Grant ECCS 1710558. Jonas Hartwig is supported by theSimons Foundation under Grant 637600. The authors would like to thank Dr. F. Albertini for helpful discussionson the topic of this paper. The authors would also like to thank an anonymous referee for a very careful reading,constructive criticism and several suggestions concerning a first version of this paper.
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Physical Review A , , 042309 (2015) Appendix A: Proofs of Formula (38) and of Formula (51)
Consider a Lie algebra R which has a basis B := { E j } which is invariant, as a set, under the action of the group G , i.e., if E ∈ B , gEg − ∈ B , ∀ g ∈ G . Then we can derive a basis for R G : Let O the set of orbits of G in B underthe above action. Proposition 5.2.
The set of elements { (cid:88) E j ∈ O E j | O ∈ O} , (58)is a basis of R G . In particular, the dimension of R G is equal to the number of orbits in B under the above describedaction of G on B . 22 roof. Since the E j ∈ B form a basis and the orbits are disjoint, then elements (cid:80) E j ∈ O E j in (58) for differentorbits O are linearly independent. Moreover write F ∈ R G as F = (cid:80) O ∈O F O where O is the set of orbits and F O is a linear combination of elements in B in the orbit O . Since gF g − = F for each g ∈ G , and each orbit isinvariant, we have gF g − = (cid:88) O ∈O gF O g − = F = (cid:88) O ∈O F O , which implies that, for every orbit O , and every g ∈ G , gF O g − = F O . Write F O = (cid:80) j α j E j where E j are theelements in the basis B which also belong to the orbit O , and for some coefficients α j . Fix j and k and a g ∈ G sothat g maps E j to E k . Such a g always exists because, by definition, the action of G is transitive on its orbits. Byimposing gF O g − = F O , using the fact that the map associated with g is a bijection from the orbit to itself, wefind that α j = α k . Since, this is valid for arbitrary j and k , we find that F O must be proportional to the elements (cid:80) E j ∈ O E j in the set (58).According to the proposition, the dimension of R G can be calculated using the Burnside’s orbit counting theorem (see, e.g., [21]), orbits = 1 | G | (cid:88) g ∈ G | Fix g | , (59)where Fix g denotes the set of elements fixed by g , in B . Proof.
By Proposition 5.2 we have dim u C n (2 n ) = C n on the set of all words of length n in thefour symbols , σ x , σ y , σ z .Recall Burnside counting theorem (59) which applied to our case gives: | C n | (cid:88) g ∈ C n | Fix g | . (61)The cyclic group C n = (cid:104) Z (cid:105) has a unique subgroup H m of order m for every positive divisor m of n , namely H m = (cid:104) Z n/m (cid:105) . Since every element g of C n generates some subgroup, we can partition C n into subsets corresponding towhich subgroup they generate. Then we get n (cid:88) m | n (cid:88) g ∈ C n (cid:104) g (cid:105) = H m | Fix g | , (62)where (cid:80) m | n means we sum over all positive integers m which divide n . Next we use the fact that a word is fixedby g if and only if it is fixed by the cyclic subgroup (cid:104) g (cid:105) . Thus we get from (62) n (cid:88) m | n (cid:88) g ∈ C n (cid:104) g (cid:105) = H m | Fix H m | . (63)Now recall that any cyclic group has many possible generators. In particular if g generates a group G of order m , g a generates G if and only if gcd( a, m ) = 1. Applying this to G = H m , which is cyclic of order m , ( Z nm ) a generates H m if and only if gcd( a, m ) = 1. The Euler’s totient function φ ( m ) counts the number of positive integers a lessthan or equal to m having greatest common divisor 1 with m . Therefore H m has φ ( m ) generators. This meansthat we can rewrite (63) as follows: n (cid:88) m | n | Fix H m | · φ ( m ) (64) We use the standard convention in group theory denoting by (cid:104) F , F , ..., F s (cid:105) the group generated by the set { F , F , ..., F s } . m is a positive integer that divides n then the number of words of length n in 4 letters that are fixed by H m (equivalently, by Z n/m ) is 4 n/m because such words are uniquely determined by the first n/m positions, which canbe arbitrarily chosen. This gives us the formula we wanted to show u C n (2 n ) = 1 n (cid:88) m | n n/m φ ( m ) . With the same steps as in the previous proof applied to X k rather then the whole set of words we arrive at (cf.,formula (64)) | X k /C n | = 1 n (cid:88) m | n | Fix H m | · φ ( m ) , (65)where now the set fixed by H m , Fix H m is considered in X k rather than in the space of all 2 n binary words. Recallthat H m is the subgroup generated by Z n/m . A word a in X k is fixed by H m if and only if Z n/m ( a ) = a . Thisin turn holds if and only n/m is a multiple of the period T of a . Therefore the words in Fix H m have period T such that n/T divides k and m divides n/T . Their number by definition is w ( n, m, k ). Moreover in the sum (65) m has to divide n/T and therefore n , and n/T has to divide k , so that m also has to divide k . Therefore thenonzero terms are obtained for m at most equal to the greatest common divisor of n and k , i.e., gcd( n, kn, k