Rotational-permutational dual-pairing and long-lived spin order

Quantum systems in contact with a thermal environment experience coherent and incoherent dynamics. These drive the system back towards thermal equilibrium after an initial perturbation. The relaxation process involves the reorganisation of spin state populations and the decay of spin state coherences. In general individual populations and coherences may exhibit different relaxation time constants. Particular spin configurations may exhibit exceptionally long relaxation time constants. Such spin configurations are known as long-lived spin order. The existence of long-lived spin order is a direct consequence of the symmetries of the system. For nuclear spin systems rotational and permutational symmetries are of fundamental importance. Based on the Schur-Weyl duality theorem we describe a theoretical framework for the study of rotational and permutational dual-symmetries in the context of longlived spin order. Making use of the proposed formalism we derive exact bounds on the number on long-lived spin populations and coherences for systems exhibiting rotational-permutational dual-symmetries.

Spin configurations with exceptional lifetimes arise if the system exhibits a certain degree of symmetry. The application of symmetry arguments for the description of NMR experiments has a longstanding history 19,[32][33][34][35][36][37][38][39][40][41][42][43] . It is therefore not too surprising that the theory of long-lived nuclear spin order is most naturally formulated within a group theoretical framework. Here the collection of symmetry operations that leave the system invariant form a mathematical group G [44][45][46][47] . The symmetries of the system may then be studied by making use of fundamental results from group theory [4][5][6][14][15][16][17][18][19]48 . Most importantly the presence of symmetries implies the existence of selection rules which may be determined by the generalised Wigner-Eckart-Theorem 46,47 . The underlying Hilbert space H then decomposes into a direct sum of irreducible representation Γ j of the group G. Groups of special interest for NMR applications are the three-dimensional rotation group SO(3) and the symmetric group S n 3-5, [14][15][16][17][18][19]36,38,39 .
Exceptional relaxation time constants follow from symmetry induced selection rules that either forbid or greatly suppress transitions between different symmetry manifolds by coherent and incoherent mechanisms 3-8, [14][15][16][17][18] . As a consequence information across different symmetry manifolds may be stored for an extended period of time. Meaningful long-lived spin order may then be defined as the population difference across distinct symmetry manifolds 3, [14][15][16][17][18]48 . In the case of two coupled spin-1/2 particles for example a long-lived state may be defined as the population imbalance between the anti-symmetric singlet and symmetric triplet manifold 12 .
In more complicated spin systems an intuitive argument to determine the number of longlived spin order becomes increasingly difficult. Some long-lived spin operators for particular spin systems have been reported previously 4,5,[14][15][16][17][18][49][50][51][52][53][54][55] . Recently references 16 and 56 have given a lower bound on the number of long-lived spin populations based on group theoretical arguments. Long-lived spin populations may be constructed from a set of projection operators P Γ j that project onto the different symmetry manifolds Γ j of a group G. The number of non-trivial long-lived spin operators is bounded by the relation 16 where N Γ represents the number of irreducible representations of G.
This bound however displays some issues. Consider the following (apparent) paradox: The three protons of a methyl rotor constitute a system of three identical spin-1/2 particles. In a typical solution state NMR experiment the symmetry of the methyl group is commonly described by the alternating group of order three A 3 14-17,37 . The group A 3 may be characterised by three irreducible representations: {A, E a , E b } leading to N Γ = 3. It is however straightforward to show that the symmetry of the spin Hamiltonian for the methyl protons is equally well described by the symmetric group of order three S 3 . The group S 3 is similarly characterised by three irreducible representations: {A 1 , A 2 , E}. Spin functions for three identical spin-1/2 particles however are not capable of supporting the A 2 irreducible representation [44][45][46][47] . This reduces the number of available irreducible representations by one, so that N Γ = 2. As a consequence the number of long-lived spin operators predicted by characterisation according to A 3 or S 3 symmetry are not the same. This has lead to some confusion regarding which symmetry group of the Hamiltonian should be used to describe the system.
In this paper we address and resolve this issue by treating rotational and permutational symmetries on equal footing. This is done by exploiting the dual-pairing for the irreducible representations of the unitary group U(n) and the symmetric group S n . The intimate relationship between these two groups is described by the Schur-Weyl duality theorem 57,58 .
The Schur-Weyl duality theorem gives a concrete relation between the irreducible representations of S n and U(n). Since the three-dimensional rotation group is a subgroup of the unitary group U(n) the duality principle provides a connection between the irreducible representations of S n and SO (3). This may be done by utilising branching rules for irreducible representations for the subgroup chain SO(3) ⊂ U(n). This way we will be able to state a new lower bound on the number of long-lived spin operators including spin populations and coherences.
For practical applications a simple approach is outlined to construct basis sets that possess dual-symmetry for any two commuting groups G a and G b and in particular the case G a = S n and G b = SO(3) is treated. Such basis sets may be used to verify the presented results and simplify explicit calculations of relaxation dynamics. We illustrate this by reexamining the long-lived states of a methyl rotor under dual-selection rules. A detailed application to practical NMR examples however is beyond the scope of this paper and deferred to a future publication currently under preparation.

II. BACKGROUND
A short overview of key concepts and necessary terminology is given in order to facilitate the following discussion. The required group theoretical concepts are presented in full detail in references 44, 46 and 47.

A. Groups and Subgroups
Consider a set of elements G equipped with a composition law ( * ) for any two elements g i , g j ∈ G. The composition law is often abbreviated by group product or simply product. If (G, * ) satisfies the axioms: • The product g i * g j is an element of G for all elements g i , g j ∈ G.
• The product of the group elements g i * g j * g k is associative.
• The identity element denoted by e is an element of the group G.
• For every element g i there exists the corresponding inverse element g −1 i ∈ G so that the set G together with ( * ) forms a group [44][45][46][47] .
If the group G contains a subset A so that A fulfills the group axioms, the subset A is called a subgroup of G. Subgroups are commonly denoted by A ⊂ G. In general A may contain additional subgroups leading to a so-called subgroup chain, N is the length of the subgroup chain 47 .

B. Symmetric group
The symmetric group naturally arises in physical systems consisting of a set of identical particles. Consider a set of N elements Ω N = {1, 2, . . . , N }. A permutation is as a rearrangement of elements in Ω N in a one-to-one manner 44 . For example if Ω N is given by {1, 2, 3} a permutation may be defined as follows: This sequence of operations is often abbreviated by the so-called cycle notation The collection of all such permutations on the set Ω N satisfies the group axioms and forms a group G of order N !. The resulting group is called the symmetric group of order N ! and denoted by S n .
A simple example is the symmetric group on a set of two objects denoted by S 2 . The 2! = 2 elements of S 2 are given by where the permutation (1)(2) is the identity element in S 2 and in general (1)(2) . . . (n) = e in S n . The symmetric group on a set of three objects is denoted by S 3 and has 3! = 6 elements S 3 = {e, (12), (23), (13), (123), (132)}.
The symmetric group S 3 is the first symmetric group that contains a non-trivial subgroup.
This subgroup is the set of all even permutations In general the set of all even permutations on a set of order N is called the alternating group A n ⊂ S n . The alternating group A 3 is commonly used to analyse the dynamics and describe symmetries of methyl group rotors [14][15][16]18,37,59 . In the NMR literature however it is common practice to refer to the groups S 3 and A 3 by the crystallographic point groups C 3v and C 3 . This is possible since the groups S 3 ∼ = C 3v and A 3 ∼ = C 3 are isomorphic to each other.

C. Dynamic groups
The unitary group U(n) consists of all n × n matrices that satisfy the condition so that U −1 equals its conjugate transpose or adjoint denoted by U † . The unitary group U (n) forms a connected continuous group of n 2 parameters 44,46 . Group elements of U(n) may be generated as follows with α j being a real number and E j a n × n hermitian matrix that belongs to corresponding Lie algebra u(n).
An important subgroup of the unitary group is the special unitary group SU(n) ⊂ U(n).
Elements of SU(n) fulfill the additional constraint det(U ) = 1. The group elements of SU(n) are generated by n 2 − 1 traceless hermitian matrices 44,46 su = exp −i The generators of the Lie algebra su(n) may be chosen to be polynomials in the Cartesian angular momentum operators {I x , I y , I z }, which naturally arise in the definition of a spin Hamiltonian 60-63 .

D. Conjugacy classes
The elements of a group G may be partitioned into conjugacy classes. Two elements g i and g j of G are conjugate to each other if the following relation holds The set of all elements conjugate to each other is called a conjugacy class and g i is called the representative of the class.
For the symmetric group elements with the same cycle structure form a conjugacy class 44 .
For example the elements (123) and (132) and the elements (12)(34) and (13) The possible cycle structures for S n may be determined by considering all integer partitions λ of n 72 . The symbol λ n will be reserved to indicate that λ partitions n. An integer partition of n represents n as a sum of one or more positive integers. The integer partitions of 5 for example are given by Thus there is a natural identification between conjugacy classes of S n and the integer partitions of n.
The conjugacy classes for matrix groups such as U(n), SU(n) and SO(3) are formed by all matrices that are related to each other by a similarity transformation Since the trace of a matrix is preserved under a similarity transformation the conjugacy classes are given by all matrices with the same trace. For the rotation group the conjugacy classes are formed by all rotations with the same effective rotation angle θ. Conjugate rotations then simply correspond to the same rotation as viewed from different reference frames.

E. Irreducible representations
For certain applications it is convenient to generate a representation D(g) for the group elements g ∈ G. This is done by mapping the group elements onto (d × d) dimensional matrices. The group composition law then translates into matrix multiplication. If the mapping is additionally one-to-one the representation is said to be faithful.
For groups considered here there always exists a similarity transformation V that blockdiagonalises the representations D(g) simultaneously so that all D(g) ∀g ∈ G admit a direct The Depending on the underlying Hilbert space H some irreducible representation may appear multiple times and some may not appear at all. The multiplicity of a given irreducible representation is denoted by m Γ j . The dimension of the Hilbert space H may then be expressed as follows The basis set that spans the irreducible representations of a given group G is called a symmetry adapted basis B G . The members of B G may be specified by a set of group labels 46,47 The first label Γ j specifies the irreducible representation the state belongs to. The second label γ distinguishes between basis elements belonging to the same irreducible representation with d Γ j > 1. In this work we will refer to this label as the group projection number. The label µ is necessary for m Γ j > 1 to uniquely address each basis element.
This notation is chosen to resemble the labelling of angular momentum states |I, m with the addition of a multiplicity label.

F. Characters
The irreducible representations of a group may be classified according to their characters.
The character χ of a representation is given by the trace of the corresponding matrix From equation 17 it follows that the character χ(g) may be expressed as follows The characters χ Γ j (g) are called irreducible characters of g ∈ G. The irreducible characters are orthogonal to each other with respect to the following complex inner product This is a consequence of the Schur orthogonality relations 44-47 for the matrix elements [D Γ j (g)] mn of the irreducible representation Γ j . An irreducible representation Γ j is therefore uniquely specified by its set of characters.
The characters are conveniently collected into a character table. An example of a typical character table is given in table I.
The first column runs through all irreducible representations of the group. The following columns are labelled by the group elements of G.

Symmetric group
According to section II E the number of irreducible representations equals the number of conjugacy classes. For S n the conjugacy classes are in a one-to-one correspondence with the integer partition of n (see section II D). As a result the irreducible representations of S n may also be labelled by integer partitions.
It is common to represent the irreducible representation of S n by the integer partitions and the classes by a cycle representative. In the case of S 3 this results in a character The assignment of a given integer partition to an irreducible representations follows particular symmetry arguments that may be found in reference 44.

Three-dimensional rotation group
The character table for SO(3) may be determined by considering solutions of the threedimensional Laplace equation The solutions to this equation are spherical harmonics Y lm (θ, φ) 46 The irreducible representations of SO(3) are thus distinguished by the quantum number l.
The spherical harmonics Y lm (θ, φ) are additionally labelled by a quantum number m to distinguish elements with the same quantum number l but different transformation properties under the two-dimensional rotation group SO(2).

G. Direct product groups
Given two groups G a and G b one may define the direct product group G a ⊗ G b by introducing the following element-wise operation [44][45][46][47] where ( * a ) and ( * b ) are the group products for G a and G b , respectively. The direct product group G a ⊗G b contains subgroups G a and G b which are isomorphic to G a ∼ = G a and G b ∼ = G b .
The isomorphism may be constructed as follows so that the two groups G a and G b commute and every element of G a ⊗ G b may be represented as a product of an element from G a and If both groups G a and G b are realised in terms of representations D(g a ) and D(g b ) the direct product group may be constructed by forming the Kronecker product of the matrices D(g a ) and D(g b ) We further assume that we are dealing with an ensemble of such systems dissolved in a liquid. The state may then be described by a density operator ρ where the overbar indicates an average across the spin-ensemble. The density operator ρ is an element of the space of linear operators on H denoted by L(H) In the NMR jargon L(H) is known as Liouville space 61 . Making use of superoperator notation the Liouville-von-Neuman equation governing the dynamics of the ensemble may be expressed as follows 73 d dt |ρ(t)) = (L coh +R lab )|ρ(t)), withL coh being the coherent Liouvillian of the system,R lab the generator of relaxation expressed in the laboratory frame and |Q) denotes a state in L(H). The coherent Liouvillian is assumed to be the same for each member of the ensemble and is given by the commutation The assumption of an ensemble of N identical spins dissolved in an isotropic liquid allows us to express the coherent Hamiltonian as follows [61][62][63] with I z being the total z-angular momentum operator. The interaction between the spins and the magnetic field is characterised by a single Larmor frequency ω 0 . The second term describes the mutual scalar coupling between pairs of spins with the same coupling strength J.
The relaxation superoperatorR lab may be constructed from the fluctuating part of the spin LiouvillianL fluc (t)L The fluctuating part is in general different for each member of the ensemble at given point it time. For conventional spin systems the relaxation superoperatorR lab takes the form of a double commutation superoperator 61,62 whereL fluc (t) represents the fluctuating part expressed in the interaction frame of the coherent LiouvillianL The relaxation superoperator for spin systems deviating strongly from thermal equilibrium should be formulated within the Lindblad formalism 74,75 . The symmetry properties of the relaxation superoperator do not depend upon this choice and we will avoid this complication here.

B. Rotational symmetries
An operator Q possesses rotational symmetry if it is invariant under conjugation by Being invariant under rotations implies that Q preserves the total angular momentum I and z-angular momentum m of a state in H.

The coherent Hamiltonian H coh is in general not invariant under SO(3) but commutes
with the total angular momentum operator I 2 and the total z-angular momentum operator where I 2 and I z are defined as follows: As a consequence the coherent Hamiltonian also preserves total angular momentum I and zangular momentum m, and eigenstates of H coh may be chosen to be total angular momentum states |I, m The set of all angular momentum states |I, m forms a symmetry adapted basis B SO (3) for Similarly, the total angular momentum superoperatorÎ 2 and total z-angular momentum superoperatorÎ z in L(H) are given by the expressions beloŵ The eigenoperators of (Î 2 ,Î z ) are given by irreducible spherical tensor operators (ISTO) 60 .
Each ISTO may be labelled by two quantum numbers (k, m) which reflect their symmetry under SO(3) and SO(2) operations The set of all spherical tensor |T km ) forms a symmetry adapted basisB SO(3) for rotations in Consider the coherent Liouvillian resulting from the Hamiltonian in equation 34. It is straightforward to show thatL coh commutes withÎ 2 andÎ z The coherent LiouvillianL coh thus preserves the two quantum numbers (k, m) and its representations may be decomposed according to the irreducible representations of the rotation group SO (3).
The relaxation superoperatorR lab on the other hand possesses no symmetry in general.
But the combination of dominant Zeeman interactions and weak relaxation processes driven by rotational Brownian motion imposes symmetry onto the relaxation superoperator 64,65 .
TypicallyR lab is well approximated by a secularised relaxation superoperatorR which possesses at least SO(2) symmetry so thatR preserves the z-angular momentum of any ISTO. Without any further approximations or assumptionsR may not be decomposed into a sum of irreducible representations of the rotation group SO(3).

C. Permutational symmetries
A system of N identical spins possesses spin permutation symmetry. The action of a spin permutation P σ onto a direct product state |ψ 1 ψ 2 . . . ψ N may be expressed as shown below For instance, the permutation P 12 acting on the state |ψ 1 ψ 2 ψ 3 leads to the following The action of a spin permutation P σ onto an operator Q is given by conjugation of Q by P σ When an operator is expressed in terms of Cartesian angular momentum operators for example, a spin permutation amounts to a permutation of the indices Consider the coherent Hamiltonian of equation 34. Since the nuclear Larmor frequency ω 0 and the mutual scalar coupling constant J is the same for all members of the system, the coherent Hamiltonian is invariant under the symmetric group S n Similarly, the coherent LiouvillianL coh is invariant under the set of spin permutation super- that generate a representation of the group S n on L(H) As a consequence the representations of H coh andL coh may be decomposed according to irreducible representations of S n . And again without further approximationsR does not possess any spin permutation symmetry.

D. Schur-Weyl duality
Sections III B and III C showed that the coherent Liouvillian and Hamiltonian given by equations 33 and 34 are invariant under group elements S n and preserve total and z-angular momentum. This gives rise to the concept of a dual-symmetry 57,58 .

Consider a Hamiltonian
H or LiouvillianL that commutes with two groups G a and G b with the additional assumption that G a and G b commute It is easy to check that that this condition is fulfilled for spin rotations and permutations.
In this case one speaks of a dual-symmetry and there exists a set of basis states and basis operators that transforms irreducibly under G a and G b simultaneously. Similar to equation 19 the basis elements may then be specified by a set of two group labels 47 In general there is no special relationship between the irreducible representations of G a and G b . An exception exists for the case that G a = U(n) and G b = S n . The connection between irreducible representations of U(n) and S n is described by the Schur-Weyl duality theorem 57,58 . Following reference 58, a simplified version of this theorem states that the direct product representation of U(n) and S n may be used to decompose H according to the irreducible representations {λ} of U(n) and [λ] of S n as follows where l(λ) indicates the length of a partition and we denote the dual pairing by (•). There is thus a unique pairing between irreducible representations of U(n) and S n identified by a common partition λ.
The Schur-Weyl duality further ensures (not captured by equation 58) that the multiplicity m {λ} of an irreducible representation of U(n) is the same as the dimensionality d [λ] of the corresponding irreducible representation of S n and vice versa.
Since the rotation group SO (3)  The coupled angular momentum basis for a system of two spin-1/2 particles consists of three triplet states |T 1m with I = 1 and a singlet state |S 00 with I = 0 These states have definite permutation symmetry under the spin permutation P 12 with the singlet state being anti-symmetric and the triplet-state being symmetric. The same information is encoded by the Schur-Weyl duality theorem. According to equation 58 it is only necessary to consider integer partitions with l(λ) ≤ 2. For two particles the possible partition are given by The branching rule for the SO(3) ⊂ U (2) subgroup chain follows the simple rule I = This leads to the following dual pairing of irreducible representations of SO(3) ⊂ U (2) and Equation 63 agrees with equations 59 and 60 if we identify [2] with the symmetric and [11] with the anti-symmetric irreducible representation of S 2 . And as indicated in  Similarly for three identical spin-1/2 particles the following partitions are of relevance Repeating the argument above one may decompose H as follows The Schur-Weyl duality thus explains the absence of the [111] or A 2 irreducible representation of S 3 for the coupling of three spin-1/2 particles [14][15][16][17]59 . With the help of the S 3 character table (see table II) it is straightforward to construct the dimensionality and multiplicity pairings as shown in table V.
For systems with I > 1/2 the branching rules for the SO(3) ⊂ U (2I + 1) subgroup chain are not unique. In these cases the branching rules can be built up recursively based on ideas by Jahn 76 . We illustrate the Schur-Weyl duality for spins with I > 1/2 by considering three coupled spin-1 particles. Such systems are interesting in the context of long-lived spin operators for deuterated methyl rotors 18,77 . For spin-1 particles it is necessary to consider partitions up to length three: The dual pairings for SO(3) ⊂ U (3) are given by the relations below Since the irreducible representations from U (3) do not remain irreducible upon restriction to SO(3) the dual-pairings between SO(3) ⊂ U (3) and S 3 are not unique.
The multiplicities of the irreducible representations may be determined by realising that every individual pairing has to fulfill the multiplicity-dimensionality pattern. This leads to a simple set of rules.
• (2) Multiply the SO (3) irreducible representation by the dimensionality of the S n irreducible representation and vice versa.
For the current example this reduces to the following sequence of steps: The last line of equation 68 has been summarised in table III D to illustrate the individual dual-pairings. While it is straightforward to determine the dimensionality of an irreducible representation D I , it is not straightforward to calculate the dimensionality of an irreducible representation . Since the following discussion will focus on spin systems with I = 1/2 and I = 1 we provide expressions for the dimensionality of A derivation of these is given in appendix A 1.
E. Long-lived spin operators

Permutational symmetry
Long-lived spin populations and long-lived spin coherences possess exceptional relaxation time constants. This indicates that they are either exact or approximate members of the nullspace ofR (strictly speaking one should consider the nullspace ofL coh +R, but this distinction will be inconsequential for the following discussion) 48 . The elements of the nullspace Null(R) may be constructed by imposing symmetries onR. Practically this is achieved by synthesising molecules with a high degree of local symmetry and rigid molecular structure [78][79][80] .
In the NMR literature it is often assumed that long-lived spin operators are encountered ifR is invariant under the permutation group of H coh (or one of its subgroups) [14][15][16] . But this conclusion is incorrect. The relaxation superoperator has to obey a stricter symmetry principle. For long-lived spin operators to existR has to be invariant under the following direct product group The elements of S 2 n are given by generalised spin permutation superoperatorsP τ These operate on H and its dual H * , respectively.
To see whyR has to be invariant under S 2 n , we first consider a system without any symmetry. In this case the only member of Null(R) is the sum of all populations |1) The identity operator may be decomposed by making use of the projectors P [λ] onto the different symmetry manifolds of S n on H 44-47 Equation 72 may then be expressed as shown beloŵ where we have defined the following projection superoperator Assuming thatR commutes with every projectorP [λ] equation 74 reduces to the followinĝ The projectorsP [λ] however may not be constructed from the set which generates a representation of S n on L(H). Instead they are constructed from the elements of the group S 2 n . This implies thatR has to be at least S 2 n symmetric. The physical interpretation for this is straightforward. Consider for example a system dominated by dipolar relaxation. The dipolar relaxation superoperator may be expressed as follows 61,62 :R =R + +R × , withR + being auto-correlation terms andR × being cross-correlation terms. The commutation superoperatorsT ij 2m are constructed by coupling rank one ISTO of spins i and j into a rank two ISTO. The relaxation rate constants b Instead of acting onto the superoperators the permutations may alternatively act on the relaxation rate constantsP This way it is easy to see that S n operations on L(H) permute auto-correlation terms and cross-correlation terms, but never mix auto-and cross-correlation terms for mixed m. S n symmetry therefore describes the symmetry ofR + andR × separately. For exact long-lived spin operators to exist however, it is necessary thatR remains invariant even if auto-and cross-correlation terms are exchanged. Such symmetries are described by S 2 n and not by S n . Making use of the considerations above we have the following result regarding the nullspace of a relaxation superoperatorR.
Theorem III.1. The nullspace Null(R S 2 n ) of an S 2 n symmetric relaxation superoperatorR S 2 n is given by the set of generalised group projection operators P If considered as states in L(H) the action ofR S 2 n onto |P γδ ) may be expressed as followŝ where the second line is a result of the S 2 n symmetry ofR S 2 n . The set of group projectors γδ ) therefore spans the nullspace ofR S 2 A bound on the number of long-lived spin operators may be determined by expressing the P

[λ]
γδ in terms of the symmetry adapted basis states B Sn The number of long-lived spin operators is then bounded by the following relations: Corollary III.1.1. For a S 2 n symmetric relaxation superoperatorR S 2 n the number of longlived spin populations N P LLS is bounded by Corollary III.1.2. For a S 2 n symmetric relaxation superoperatorR S 2 n the number of longlived spin coherences N C LLS is bounded by and S 3 symmetry are N P LLS = 3 and N P LLS = 2, respectively. By taking the dimensionality of the irreducible representations into account both groups A 3 and S 3 lead to identical results, namely N P LLS = 3. This may be seen as follows. The coherent Hamiltonian for a methyl-rotor is S 3 and A 3 symmetric. Since A 3 is a subgroup of S 3 it is always possible to choose a symmetry adapted basis that remains irreducible for the subgroup chain A 3 ⊂ S 3 . In table VII we have listed the branching rules for the A 3 ⊂ S 3 subgroup chain. Upon restriction from S 3 to A 3 the two-dimensional E subspace splits into two one-dimensional subspaces E a and E b . As a consequence the overall number of long-lived populations remains constant. Consider now the number of long-lived spin coherences. It is clear that the prediction of long-lived coherences differs for the groups S 3 and A 3 . For the group S 3 one finds N C LLS = 2 since the irreducible representation E is two-dimensional. But for the group A 3 one finds N C LLS = 0. This simple example shows that the bigger group, S 3 in this case, carries more information than A 3 , so that in general one should always use the maximal symmetry group of the relaxation superoperatorR.
Besides the existence of long-lived spin operators, the S 2 n symmetry imposes a block structure onto the matrix representation ofR S 2 n when expressed in the outer tensor basis of symmetry adapted basis states B Sn where we have introduced the following short-hand notation A general matrix element ofR S 2 n may then be expressed as follows The last line may be simplified by making use of the Schur-orthogonality relations (see In a second step one may replaceR S 2 n byP σ, † eR S 2 nP σ e and perform the same steps again. A generic matrix element is then constrained by the following relation:

F. Rotational symmetries
So far we have only considered permutational symmetries of the relaxation superoperator.
We will continue assuming thatR is S 2 n symmetric but additionally consider rotational symmetries.
The treatment of rotational symmetries for a relaxation superoperator requires some care.
In principalR S 2 n may possess "two types" of rotational symmetry. To see this a generic S 2 n symmetric relaxation superoperatorR S 2 n originating from rotational diffusion processes where the superoperatorsÂ are derived from linear and the superoperatorsB from bilinear spin operators 61,62 . S 2 n permutation symmetry implies that the relaxation rate constants a km and b km are independent of (i, j, p, q) for fixed (k, m). This enables the definition of effective commutation superoperators as shown below This is often the case for relaxation in the fast motion limit, but we will not make this assumption here 61 In this case the matrix elements of equation 101 obey dual-selection rules reflecting their rotational and permutational symmetry.

Dual-selection rules may be derived by introducing dual-basis states into equation 101.
The first contribution is subjected to the following selection rules where j runs over all total angular momentum values that may pair up with the irreducible representation D [λ] or D [κ] . The symbol ∆(j 1 , j 2 , j 3 ) describes the triangular conditions the angular momenta have to satisfy. It has the following properties 4,5,44-47 The third and fourth contributions to the matrix elements ofR S 2 n are significantly simpler and are constrained by the same selection rules δ I 1 I 3 ∆(I 2 , k, j)∆(I 4 , k, j) − ∆(I 1 , k, I 3 )∆(I 2 , k, I 4 ) , where we have made us of equation 93 to introduce the missing group selection rules.
The selection rules of equation 107 are readily generalised for the case thatR ∆(I 2 , k, j) − ∆(I 1 , k, I 3 )∆(I 2 , k, I 4 ) . (108) When expressed in the outer tensor product basis of dual-symmetry adapted basis states For spin-1/2 systems it is therefore sufficient to consider diagonal triangular conditions.
To analyse the implications of such dual-selection rules it is advantageous to treat spin-1/2 systems with an even and odd number of particles separately. As it turns out these two cases will behave differently under rotational-permutational dual-symmetry.
A. Odd number of spins  This idea may be generalised to any odd system of N identical spins with I = 1/2. As indicated by equation 112 any odd number of systems contains the representation D 1/2 and according to equation 111 the D 1/2 representation pairs with the integer partition [ 1 2 (N + 1) 1 2 (N + 1) − 1]. For example for 3, 5, and 7 spins the pairings D 1/2 • D [21] , D 1/2 • D [32] , D 1/2 •D [43] are present. The total number of long-lived spin operators may then be calculated by making use of the dimensionality function d λ,2 for partitions with l(λ) ≤ 2 (see equation 69) The number of long-lived spin populations and spin coherences may then be separated as indicated below where m 1 and m 2 may take the values +1/2 and −1/2.

V. SYSTEMS OF SPIN-1 PARTICLES
In contrast to spin-1/2 systems a general discussion for spin-1 particles becomes difficult.
This is due to the fact that the rotational-permutational dual-pairing is not unique anymore.
As a consequence one has to consider all possible triangular conditions occurring in equation 107.
To illustrate the general strategy we consider three identical spin-1 particles These could for example represent a deuterated methyl rotor. According to equation 67 such systems are able to support irreducible representations [3], [21] and . The matrix representation ofR S 2 n takes the following form   γδ however contains contributions from spin states with I = 1 and I = 3. This may be seen by expressing P [3] γδ in terms of symmetry adapted states with P [λ]I γδ being the projector onto the [λ] subspace with total angular momentum I. The triangular conditions above then imply that both P The projector P [3] γδ is thus split into two long-lived spin operators. Finally, according to equation 124 the diagonal symmetry block ([21] [21], [21][21]) is not subjected to any rotational selection rules. The number of long-lived spin operators taking rotational symmetries into account increases from N LLS ≥ 6 to N LLS ≥ 1 + 2 + 4 ≥ 7.
A comparison with numerical results however shows that this number is greatly exceeded.
where we have suppressed the group and z-angular momentum projection number. As shown in appendix A 4 the effective double commutation superoperatorÂ The vanishing commutator leads to two additional symmetry constraints. A relaxation superoperatorR S 2 n with k = 1 being the only active relaxation mechanism has to preserve both total angular momentum quantum numbers I 1 and I 2 of an operator |[λ], I 1 ; [κ], I 2 ) This type of symmetry has implications for diagonal symmetry blocks that are not uniquely associated with a single total angular momentum quantum number. For example, the symmetry block ([21] [21], [21][21]) associated with angular momentum states with I = 1 and I = 2 is not subjected to triangular conditions, but is subjected to this additional type of symmetry. In particular this means, similar to the projector P [3] γδ , the projector P [21] γδ may be decomposed according to its total angular moment components At this point we are not able to systematically identify these operators as they do not result from further splittings of the projection operators.

VI. DUAL-SYMMETRY BASIS SETS
For practical applications we will briefly outline a method to construct dual-symmetry adapted basis states. A thorough discussion of the necessary group theoretical tools, including proofs, may be found in reference 47.
For a given group G 0 define a class operator C G 0 as follows with h being an arbitrary class representative. The class operator C G 0 commutes with every and each class operator C G 0 possesses a complete set of eigenstates {|φ j } In agreement with quantum mechanical convention the eigenstates may be labelled according The eigenvalues {ζ j G 0 } are related to the irreducible characters of G 0 . If the group G 0 is abelian the set of eigenvalues {|ζ 1 G 0 , |ζ 2 G 0 , . . . , |ζ N H G 0 } will be non-degenerate and every eigenstate may be uniquely specified by a single eigenvalue. In this case the set of eigenstates In general the set of eigenvalues {ζ j G 0 } will be degenerate since a single eigenvalue or character is not enough to distinguish between basis states belonging to the same irreducible representation Γ j with d Γ j > 1. To uniquely address an eigenstate |φ it is necessary to assign additional quantum numbers to each state. The additional quantum numbers are provided by class operators of a given subgroup of G 0 . For example if G 1 ⊂ G 0 and C G 1 is a class operator of G 1 equation 133 indicates that C G 1 and C G 0 commute so that C G 1 and C G 0 share a common set of eigenstates |ζ j G 0 , ζ j G 1 . The idea is to add class operators of the subgroup chain G N ⊂ · · · ⊂ G 1 ⊂ G 0 until the set of eigenvalues {ζ j G 0 , ζ j G 1 , . . . , ζ j G N } uniquely labels each eigenstate |φ j . The subgroup chain is called canonical if the group G N is abelian. In this case there is no need to add any further operators and the degeneracy is completely lifted. The set of class operators To construct rotational-permutational symmetry adapted basis sets class operators of the group S n are added to the SO(3) CSCO. This is possible since every permutation operator P σ commutes with I 2 and I z .
As an example we consider a system of three identical spins. A possible choice for the CSCO is given by the following set of operators (a 1 I 2 + a 2 I z + a 3 (P 12 + P 13 + P 23 ) + a 4 P 123 )|φ j = (a 1 I(I + 1) + a 2 m + a 3 ζ j For a system of three spin-1/2 particles the solution to the matrix pencil of equation 139 leads to the following set of states

VII. AN ILLUSTRATIVE NMR EXAMPLE
Making use of the concepts presented in section IV and VI we examine the possible long-lived spin operators for protonated methyl rotors. Such systems have been analysed previously using purely rotational and permutational selection rules, but not under dualselection rules 5,15 . In the rotating frame the coherent Hamiltonian of the system is given by a symmetric scalar coupling term The system therefore displays S 3 symmetry and classifies as an odd spin-1/2 system. For such a case the arguments of section IV A indicate that dual-selection rules may lead to an increased number of long-lived spin operators.
The required degree of symmetry is a consequence of overall tumbling and internal methyl rotor dynamics. As illustrated in figure 1 the methyl rotor is assumed to be attached to a marcomolecule undergoing isotropic rotational diffusion characterised by a single correlation time τ c . The methyl rotor dynamics are described by a symmetric three-fold nearest neighbour jump model with rate constant k = (3τ r ) −1 . Focusing on dominant dipolar relaxation mechanisms, the relaxation superoperator takes the form 61,62 where b HH represents the proton-proton dipolar coupling strength, r HH the proton-proton distance, γ H the proton gyromagnetic ratio and µ 0 the magnetic constant. The commutation superoperatorsT ij 2m are constructed from irreducible spherical tensor given in appendix A 5 and C ijkl (τ ) are (classical) time correlation functions. As shown in appendix A 6 the time correlation functions (including tumbling and methyl rotation) may be expressed as a sum of S 2 3 -symmetric and S 2 3 -asymmetric contributions where C sym ijkl (τ ) = 3 10 exp (−|τ |/τ c ), and C sym ijkl (τ ) does not depend upon the indices (i, j, k, l). The relaxation superoperator may then be split into an S 2 3 -symmetric and S 2 3 -asymmetric term For sufficiently short methyl rotor correlation times (τ r τ c ), the asymmetric term may be neglected and the methyl rotor dynamics lead to a motional averaging effect. The dipolar relaxation superoperator is then approximated bŷ This situation correspond to the first row of table VIII and a total number of N LLS = 17 is expected.
The relaxation superoperator given by equation 146 has been evaluated numerically making use of the symmetry adapted basis given by equation 140. The resulting matrix representation is given in figure 2.
The matrix representation can be seen to match the pattern indicated by equation 114. As a consequence one long-lived spin operators originates from the total population of the [3] symmetry manifold with the remaining 16 long-lived spin operators being elements of the [21] symmetry manifold.

VIII. CONCLUSIONS
To summarise, we have described a purely group theoretical framework to analyse long-lived spin operators under rotational and permutational symmetries and rotationalpermutational dual-symmetries. We have considered systems for which the symmetry of the system may be described by the symmetric group S n . In this case we have shown, against common belief, that the relaxation superoperator has to be invariant under the direct product group S n ⊗ S * n and not S n for exact long-lived spin order to exist. For systems that do not display any further symmetries we have given an exact number of long-lived states and coherences. For systems with additional rotational symmetries we have derived several lower bounds on the number of long-lived spin operators. This has been achieved by first characterising the symmetries of the system according to the symmetric group S n and then including rotational selection rules by virtue of the Schur-Weyl duality theorem.
For spin-1/2 systems we have given a complete characterisation of long-lived spin operators under such rotational and permutational dual-symmetries.
An increase in the number of long-lived spin operators for spin-1/2 systems is only expected if relaxation is dominated by spherical interactions of rank two and an odd number of particles. For such systems we have given analytic expressions for the total number of long-lived spin populations and coherences. The treatment of rotational-permutational dual-symmetries of spin systems with I > 1/2 have been outlined by considering a system of three identical spin-1 particles.
The discussion for particles with higher spin quantum numbers is complicated due to the loss of uniqueness of rotational and permutation dual pairings. In general for systems with I > 1 that are subjected to relaxation mechanisms with spherical rank one or two dual-symmetries do not lead to an increase in the number of long-lived spin operators.
For systems dominated by linear relaxation mechanisms with spherical rank one we have identified additional rotational selection rules that are not captured by the Wigner-Eckart theorem. These may be used to partially explain the increase in the number of long-lived spin operators and tighten the lower bound.
Although the discussion has been mostly based on the symmetric group S n the arguments may readily be extended to other finite groups G. In certain situations the symmetric group S n may not be the appropriate group to describe the symmetries of the systems. Instead the Hamiltonian and relaxation superoperator are invariant under a smaller permutation group G. But since any finite group G is isomorphic to some subgroup of S n the rotational and permutational dual-pairing for the subgroup G may be derived by considering branching rules of irreducible representations of S n to the irreducible representations of G.
Many of the commonly encountered long-lived spin order components fall into this category. We have illustrated this by considering a protonated methyl rotor, but a separate publication is under preparation that applies the presented concepts to a broader series of practical NMR examples. Collecting all factors leads to the following product of hook lengths The dimensionality of an irreducible representation d [λ] with maximal length two is then given by the expression below This is the expression given in equation 69. Similar arguments may then be used to derive an expression for the dimensionality of irreducible representation associated with integer partitions of maximal length three.

Inner products
Given two irreducible representations D Γ a j (g a ) and D Γ b j (g a ) of the groups G a and G b the character of the resulting direct product representation is given by the following expression For the case that the two groups are identical and the group products are restricted to the diagonal case: one speaks of an inner product of group representations. In this case the product character of equation A9 reduces the the expression below The product character χ Γ j (g)χ Γ k (g) is in general not an irreducible character of G so that the representation D Γ j ×Γ k (g, g) is reducible. The reduction of the inner product follows a Clebsch-Gordan series where m jk p represents the multiplicity of the irreducible representation Γ p for the reduction Γ j × Γ k . The multiplicities m jk p may be determined from the characters of the irreducible representations which follows from the Schur-orthogonality relations given by equation 23.
Similar concepts hold for some continuous groups. In particular for the rotation group SO(3) the inner product reduction is given by the well-known angular momentum coupling rule D I 1 ×I 2 = I 1 +I 2 Such general rules for Clebsch-Gordan series are an exception however.

Dual-pairing in Liouville space
The dual-pairings indicated by equation 58 apply to basis elements of H. In certain cases it might be useful to extend rotational-permutational dual-pairings to L(H). Here rotational and permutation dual-pairings refer to the pairing of irreducible representations for rotations in L(H) and permutations generated by the set which generates a representation of S n on L(H).
According to equation 31 L(H) is given by a tensor product of H and its dual space H * .

(A22)
A quick summation over the symmetry adapted basis elements for S n or SO (3) shows that the dimensionality of the space is correct:

Commutator of commutation superoperators
A generic commutation superoperatorQ is defined as followŝ whereQ L andQ R are left and right multiplication superoperators, respectively. These act on an operator X as followsQ L X = QXQ R X = XQ.
which follows from the definition of the left and right multiplication superoperators.
The commutator between the effective commutation superoperatorÂ [N ] 1m and the left multiplication superoperator I 2 ⊗ 1 may then be expressed as follows 1m , I 2 ] ⊗ 1. (A28) The spherical tensor operators A [N ] 1m are linear combinations of the total x-, y-, and z-angular momentum operators for some coefficients c µ m . But any total angular momentum operator I µ along a particular Cartesian coordinate axis commutes with the total angular momentum operator I 2 . The effective commutation superoperatorÂ [N ] 1m then commutes with the left multiplication super- 1m , I 2 ⊗ 1] = 0.
As a result any product of the effective commutation superoperatorÂ [N ] 1m , and in particular the double commutation superoperatorÂ [N ], † 1m , commutes with I 2 ⊗ 1 and 1 ⊗ I 2, † .

Spherical Tensor Operators
Spherical tensor operators for dipolar interactions are summarised in Table X. A generic operator is denoted by T ij km . The superscript (ij) indicates angular momentum coupling of spins i and j resulting in a spherical tensor operator of total angular momentum k and z-angular momentum m.

Methyl rotor correlation functions
When expressed in the symmetry frame of the methyl rotor the dipolar interaction tensors