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Character decomposition of Potts model partition functions. II. Toroidal geometry

We extend our combinatorial approach of decomposing the partition function of the Potts model on finite two-dimensional lattices of size L x N to the case of toroidal boundary conditions. The elementary quantities in this decomposition are characters K\_{l,D} labelled by a number of bridges l=0,1,...,L and an irreducible representation D of the symmetric group S\_l. We develop an operational method of determining the amplitudes of the eigenvalues as well as some of their degeneracies.


Introduction
The Q-state Potts model on a graph G = (V, E) with vertices V and edges E can be defined geometrically through the cluster expansion of the partition function [1] Z = where n(E ′ ) and b(E ′ ) = |E ′ | are respectively the number of connected components (clusters) and the cardinality (number of links) of the edge subsets E ′ . We are interested in the case where G is a finite regular two-dimensional lattice of width L and length N, so that Z can be constructed by a transfer matrix T L propagating in the N-direction.
In a companion paper [2], we studied the case of cyclic boundary conditions (periodic in the N-direction and non-periodic in the L-direction). We decomposed Z into linear combinations of certain restricted partition functions (characters) K l (with l = 0, 1, . . . , L) in which l bridges (that is, marked non-contractible clusters) wound around the periodic lattice direction. We shall often refer to l as the level. Unlike Z itself, the K l could be written as (restricted) traces of the transfer matrix, and hence be directly related to its eigenvalues. It was thus straightforward to deduce from this decomposition the amplitudes in Z of the eigenvalues of T L .
The goal of this second part of our work is to repeat this procedure in the case of toroidal boundary conditions. This case has been a lot less studied than the cyclic case (a noticeable exception is [3]). Indeed, when the boundary conditions are toroidal, the transfer matrix (of the related six-vertex model, to be precise) does no longer commute with the generators of the quantum group U q (sl (2)). Therefore, there is no simple algebraic way of obtaining the amplitudes of eigenvalues, although some progress has been made by considering representations of the periodic Temperley-Lieb algebra (see for instance [4]). But the representations of this algebra are not all known, and therefore we choose to pursue here another approach than the algebraic one.
We use instead the combinatorial approach we developed in [2], as it is for now the only approach which can be easily extended to the toroidal case. There are however several complications due to the boundary conditions, the first of which is that the bridges can now be permuted (by exploiting the periodic L-direction). In the following this leads us to consider decomposition of Z into more elementary quantities than K l , namely characters K l,C labeled by l and a class C of permuations of the symmetric group S l . However, K l,C is not simply linked to the eigenvalues of T , and thus we will further consider its expansion over related quantities K l,D , where D labels an irreducible representation (irrep) of S l . It is K l,D which are the elementary quantities in the case of toroidal boundary conditions.
The second complication comes from the fact that, due to the planarity of the lattice, not all the permutations between bridges can be realised. It follows that the K l,D are not all independent, and so there are eigenvalue degeneracies inside and between levels.
Finally, there can be additional degeneracies because of the particular symmetry of the lattice, and even accidental degeneracies 1 . We have therefore not been able to go as far as in the cyclic case, where the amplitude of any eigenvalue in K l was given by a simple expression, depending only on l. We do however establish an operational method of determining, for any fixed (but in practice small) L, the amplitudes and degeneracies of eigenvalues in the case of a generic lattice 2 .
The structure of the article is as follows. In section 2, we define appropriate generalisations of the quantities we used in the cyclic case [2]. Then, in section 3, we decompose restricted partition functions-and as a byproduct the total partition function-into characters K l and K l,C . Finally, in section 4, we expose a method of determining the amplitudes of eigenvalues.

Preliminaries
2.1 Definition of the Z j,n1,P As in the cyclic case, the existence of a periodic boundary condition allows for non-trivial clusters (henceforth abbreviated NTC), i.e., clusters which are not homotopic to a point. 1 An example occurs for the square lattice of width L = 4, where an eigenvalue at level 1 coincides with an eigenvalue at level 2 [3], without any apparent reason. 2 By a generic lattice we understand one without mirror symmetry with respect to the transfer axis, i.e., without any accidental degeneracies. An example of a generic lattice is the triangular lattice, drawn as a square lattice with diagonals added. However, the fact that the torus has two periodic directions means that the topology of the NTC is more complicated that in the cyclic case. Indeed, each NTC belongs to a given homotopy class, which can be characterised by two coprime numbers (n 1 , n 2 ), where n 1 (resp. n 2 ) denotes the number of times the cluster percolates horizontally (resp. vertically) [5]. The fact that all clusters (non-trivial or not) are still constrained by planarity to be non-intersecting induces a convenient simplification: all NTC in a given configuration belong to the same homotopy class. For comparison, we recall that in the cyclic case the only possible homotopy class for a NTC was (n 1 , n 2 ) = (1, 0).
It is a well-known fact [6,7] that the difficulty in decomposing the Potts model partition function-or relating it to partition functions of locally equivalent models (of the six-vertex or RSOS type)-is due solely to the weighing of the NTC. Although a typical cluster configuration will of course contain trivial clusters (i.e., clusters that are homotopic to a point) with seemingly complicated topologies (e.g., trivial clusters can surround other trivial clusters, or be surrounded by trivial clusters or by NTC), we shall therefore tacitly disregard such clusters in most of the arguments that follow. Note also that the so-called degenerate clusters of Ref. [7] in the present context correspond to Consider therefore first the case of a configuration having a single NTC. For the purpose of studying its topology, we can imagine that is has been shrunk to a line that winds the two periodic directions (n 1 , n 2 ) times. In our approach we focus on the the proper-ties of the NTC along the direction of propagation of the transfer matrix T L , henceforth taken as the horizontal direction. If we imagine cutting the lattice along a vertical line, the NTC will be cut into n 1 horizontally percolating parts, which we shall call the n 1 branches of the NTC. Seen horizontally, a given NTC realises a permutation P between the vertical coordinates of its n 1 branches, as shown in Fig. 1. Up to a trivial relabelling of the vertical coordinate, the permutation P is independent of the horizontal coordinate of the (imaginary) vertical cut, and so, forms part of the topological description of the NTC. We thus describe totally the topology along the horizontal direction of a NTC by n 1 and the permutation P ∈ S n 1 .
Note that there are restrictions on the admissible permutations P . Firstly, P cannot have any proper invariant subspace, or else the corresponding NTC would in fact correspond to several distinct NTC, each having a smaller value of n 1 . For example, the case n 1 = 4 and P = (13)(24) is not admissible, as P corresponds in fact to two distinct NTC with n 1 = 2. In general, therefore, the admissible permutations P for a given n 1 are simply cyclic permutations of n 1 coordinates. Secondly, planarity implies that the different branches of a NTC cannot intersect, and so not all cyclic permutations are admissible P . Consider now the case of a configuration with several NTC. Recalling that all NTC belong to the same homotopy class, they must all be characterised by the same n 1 and P . Alternatively one can say that the branches of the different NTC are entangled.
Henceforth we denote by j the number of NTC with n 1 ≥ 1 in a given configuration.
Note in particular that, seen along the horizontal direction, configurations with no NTC and configurations with one or more NTC percolating only vertically are topologically equivalent. This is an important limitation of our approach.
Let us denote by Z j,n 1 ,P the partition function of the Potts model on an L × N torus, restricted to configurations with exactly j NTC characterised by the index n 1 ≥ 1 and the permutation P ∈ S n 1 ; if P is not admissible, or if n 1 j > L, we set Z j,n 1 ,P = 0. Further, let Z j,n 1 be the partition function restricted to configurations with j NTC of index n 1 , let Z j be the partition function restricted to configurations with j NTC percolating horizontally, and let Z be the total partition function. Obviously, we have Z j,n 1 = P ∈Sn 1 Z j,n 1 ,P , and Z j = L n 1 =1 Z j,n 1 , and Z = L j=0 Z j . In particular, Z 0 corresponds to the partition function restricted to configurations with no NTC, or with NTC percolating only vertically.
In the case of a generic lattice all the Z j,n 1 ,P are non-zero, provided that P is an admissible cyclic permutation of length n 1 , and that n 1 j ≤ L. The triangular lattice is a simple example of a generic lattice. Note however that other regular lattices may be unable to realise certain admissible P . For example, in the case of a square lattice or a honeycomb lattice, all Z j,n 1 ,P with n 1 j = L and n 1 > 1 are zero, since there is not enough "space" on the lattice to permit all NTC branches to percolate horizontally while realising a non-trivial permutation. Such non-generic lattices introduce additional difficulties in the analysis which have to be considered on a case-to-case basis. In the following, except when explicitly stated, we consider therefore the case of a generic lattice.

Structure of the transfer matrix
The construction and structure of the transfer matrix T can be taken over from the cyclic case [2]. In particular, we recall that T acts towards the right on states of connectivities between two time slices (left and right) and has a block-trigonal structure with respect to the number of bridges (connectivity components linking left and right) and a blockdiagonal structure with respect to the residual connectivity among the non-bridged points on the left time slice. As before, we denote by T l the diagonal block with a fixed number of bridges l and a trivial residual connectivity. Each eigenvalue of T is also an eigenvalue of one or more T l . In analogy with [3] we shall sometimes call T l the transfer matrix at level l. It acts on connectivity states which can be represented graphically as a partition of the L points in the right time slice with a special marking (represented as a black point) of precisely l distinct components of the partition (i.e., the components that are linked to the left time slice via a bridge).
A crucial difference with the cyclic case is that for a given partition of the right time slice, there are more possibilities for attributing the black points. Namely, a connectivity component which is not apparently accessible from the left (and thus markable) may in fact be so due to the periodic boundary conditions identifying the top and the bottom rows. This will obviously increase the dimension of the level l subspace of connectivities (for 0 < l < L). Considering for the moment the black points to be indistinguishable, we denote the corresponding dimension as n tor (L, l). It can be shown [3] that and clearly n tor (L, l) = 0 for l > L.
Suppose now that a connectivity state at level l is time evolved by a cluster configuration of index n 1 and corresponding to a permutation P . This can be represented graphically by adjoining the initial connectivity state to the left rim of the cluster configuration, as represented in Fig. 1, and reading off the final connectivity state as seen from the right rim of the cluster configuration. Evidently, the positions of the black points in the final state will be permuted with respect to their positions in the intial state, according to the permutation P . As we have seen, not all P are admissible, but it turns out to be advantageous to consider formally also the action of non-admissible permutations. This is permissible since in any case T l will have only zero matrix elements between states which are related by a non-admissible permutation. Since n tor (L, l) was just defined as the number of possible connectivity states without taking into account the possible permutations between black points, the dimension of T l is l! n tor (L, l).
Let us denote by |v l,i (where 1 ≤ i ≤ n tor (L, l)) the n tor (L, l) standard connectivity states at level l. The full space of connectivities at level l, i.e., with l distinguishable black points, can then be obtained by subjecting the |v l,i to permutations of the black points. It is obvious that T l commutes with the permutations between black points (the physical reason being that T l cannot "see" to which positions on the left time slice each bridge is attached). Therefore T l itself has a block structure in a appropriate basis.
Indeed, T l can be decomposed into T l,D where T l,D is the restriction of T l to the states transforming according to the irreducible representation (irrep) of S l corresponding to the Young diagram D. One can obtain the corresponding basis by applying the projectors p D on all the connectivity states at level l, where p D is given by Here dim(D) is the dimension of the irrep D and χ D (P ) the character of P in this irrep.
We have used the fact that all characters of S l are real. The application of all possible permutations on any given standard vector |v l,i generates a regular representation of S l , which contains therefore dim(D) representations D (each of dimension dim(D)). As there are n tor (L, l) standard vectors, the dimension of T l,D is thus [dim(D)] 2 n tor (L, l). Furthermore, using Schur's lemma, we deduce that each of its eigenvalues is (at least) dim(D) times degenerate. Therefore T l,D has (at most) dim(D) n tor (L, l) different eigenvalues, which we shall denote λ l,D,k . 3

Definition of the K l,D
We now define, as in the cyclic case [2], K l as the trace of (T l ) N . Since T l commutes with S l , we can write In distinction with the cyclic case, we cannot decompose the partition function Z over K l because of the possible permutations of black points (see below). We shall therefore resort to more elementary quantities, the K l,D , which we define as the trace of (T l,D ) N .
Since both T l and the projectors p D commute with S l , we have Obviously one has the sum being over all the irreps D of S l . Recall that in the cyclic case the amplitudes of the eigenvalues at level l are all identical. This is no longer the case, since the amplitudes depend on D as well. Indeed 3 It turns out that there are more degeneracies than warranted by this argument. The reason is that the cluster configurations cannot realise all the permutations of S l (recall our preceeding discussion), and thus some λ l,D,k with different l and/or D are in fact equal. We shall come back to this point later.
In order to decompose Z over K l,D we first introduce the auxiliary quantities the sum being over permutations P l ∈ S l belonging to the class C l . We then have So K l,P l (resp. K l,C l ) can be thought of as modified traces in which the final state differs from the initial state by the application of the permutation P l (resp. the class C l ). Note that K l,Id is simply equal to K l l! . Since the character is the same for all permutations belonging to the same class, Eqs. (5) and (3) yield a relation between K l,D and K l,P l : These relations can be inverted so as to obtain K l,C l in terms of K l,D , since the number of classes equals the number of irreps D: With the chosen normalisation, the coefficients c(D, C l ) are integer. Multiplying Eq. (10) by χ D (C ′ l ) and summing over D, and using the orthogonality We also note that 3 Decomposition of the partition function 3.1 The characters K l By generalising the working for the cyclic case, we can now obtain a decomposition of the K l in terms of the Z j,n 1 . To that end, we first determine the number of states |v l,i which are compatible with a given configuration of Z j,n 1 , i.e., the number of initial states |v l,i which are thus that the action by the given configuration produces an identical final state. The notion of compatability is illustrated in Fig. 2.
We consider first the case n 1 = 1 and suppose that the k'th NTC connects onto the points {y k }. The rules for constructing the compatible |v l,i are identical to those of the cyclic case: 1. The points y / ∈ ∪ j k=1 {y k } must be connected in the same way in |v l,i as in the cluster configuration.
2. The points {y k } within the same bridge must be connected in |v l,i .
3. One can independently choose to associate or not a black point to each of the sets {y k }. One is free to connect or not two distinct sets {y k } and {y k ′ }.
The choices mentioned in rule 3 leave n tor (j, l) possibilities for constructing a compatible |v l,i . The coefficient of Z j,1 in the decomposition of K l is therefore l! ntor(j,l) Q j , since the permutation of black points in a standard vector |v l,i allows for the construction of l! distinct states, and since the weight of the j NTC in K l is 1 instead of Q j . It follows To obtain rule 3 we have used the fact that the permutations P characterising the NTC do not have any proper invariant subspace. Note that rule 4 implies that the decomposition of K l with l ≥ 2 does not contain any of the Z j,n 1 with n 1 > 1. We therefore have simply It remains to obtain the decomposition of K 1 and K 0 . The number of standard connectivities |v l,i compatible with r big blocks is 0 for l ≥ 2 (because of rule 4); j r for l = 1 and r ≥ 1 (by rule 3 we independently choose to link up r of the j NTC, and by rule 4 the resulting big block must carry the black point); 0 for l = 1 and r = 0 (since one needs a big block to attribute the black point); and j r for l = 0. Summing over r, we finally obtain the number of compatible |v l,i : 0 for l ≥ 2; j r=1 j r = 2 j − 1 for l = 1; and j r=0 j r = 2 j for l = 0. The decomposition of K 1 reads therefore and that of K 0 is Note that the coefficients in front of Z j,n 1 do not depend on the precise value of n 1 when n 1 > 1. To simplify the notation we have defined Z 0,1 = Z 0 .

The coefficients b (l)
Since the coefficients in front of Z j,1 and Z j,n 1 >1 in Eqs. (16)-(17) are different, we cannot invert the system of relations (15)-(17) so as to obtain Z j ≡ Z j,1 + Z j,n 1 >1 in terms of the K l . It is thus precisely because of NTC with several branches contributing to Z j,n 1 >1 that the problem is more complicated than in the cyclic case.
In order to appreciate this effect, and compare with the precise results that we shall find later, let us for a moment assume that Eq. (15) were valid also for l = 0, 1. We would then obtain The coefficients b (l) play a role analogous to those denoted c (l) in the cyclic case [2]; note also that b (l) = c (l) for l ≤ 2. Chang and Schrock have developed a diagrammatic technique for obtaining the b (l) [3].
Supposing still the unconditional validity of Eq. (15), one would obtain for the full partition function This relation will be modified due to the terms Z j,n 1 >1 realising permutations of the black points, which we have here disregarded. To get things right we shall introduce l! independently of D. We shall see that the Z j,n 1 >1 will lift this degeneracy of amplitudes in a particular way, since there exists certain relations between the b (l,D) and the b (l) .

Decomposition of the K l,C l
The relations (15)-(17) were not invertible due to an insufficient number of elementary quantities K l . Let us now show how to produce a development in terms of K l,C l , i.e., taking into account the possible permutations of black points. This development turns out to be invertible.
A standard connectivity state with l black points is said to be C l -compatible with a given cluster configuration if the action of that cluster configuration on the connectivity state produces a final state that differs from the initial one just by a permutation C l of the black points. This generalises the notion of compatibility used in Sec. 3.1 to take into account the permutations of black points.
Let us first count the number of standard connectivities |v l,i which are C l -compatible with a cluster configuration contributing to Z j,n 1 ,P . For n 1 = 1, S n 1 contains only the identity element Id, and so the results of Sec. 3.1 apply: the Z j,1 contribute only to K l,Id .
We consider next a configuration contributing to Z j,n 1 ,P with n 1 > 1. The |v l,i which are C l -compatible with this configuration satisfy the same four rules as given in Sec. 3.1 for the case n 1 > 1, with the slight modification of rule 4 that the black points must be attributed to the big blocks in such a way that the final state differs from the initial one by a permutation C l . This modification makes the attribution of black points considerably more involved than was the case in Sec. 3.1. First note that not all C l are allowed. To be precise, the cycle decomposition of the allowed permutations can only contain id (the identity acting on a single black point) or P (recall that P is the permutation of coordinates realised by the branches of a single NTC). Indeed, if one attributes a black point to a big block its position remains unchanged by action of the cluster configuration, whereas if one attributes n 1 black points to the n 1 branches of one same NTC these points will be permuted by P . Furthermore, since the big blocks are automatically connected among themselves, one can at most attribute to them a single black point, and so id is contained in the cycle decomposition 0 or 1 times. Note also that the entanglement of the NTC will imply the entanglement of the structure of the allowed permutations, but this fact is of no importance here since we are only interested in C l , i.e., the classes of allowed permutations.
Denoting by n P the number of times the permutations of class C l contains P , the two types of allowed C l are: 1) those associated with permutations that only contain P , i.e., such that l = n P n 1 , and 2) those associated with permutations that contain id once, i.e., such that l = n P n 1 + 1. In the following we denote these two types as (n P , n 1 ) and (n P , n 1 ) ′ , respectively, and the corresponding K l,C l will be denoted K (n P ,n 1 ) and K (n P ,n 1 ) ′ respectively.
Let us consider the first case, which is depicted in Fig. 4. If the |v l,i have r big blocks, there are j r ways of choosing them among the j NTC, and j−r n P ways to attribute the black points. Indeed one needs to distribute l = n P n 1 black points among n P groups of n 1 non-connected blocks corresponding to the same NTC, out of a total of j − r. Since the |v l,i can contain at most j − n P big blocks, the number of C l -compatible standard connectivities is From this we infer the decomposition of K (n P ,n 1 ) : Consider next the second case. The |v l,i can still contain at most j−n P big blocks, but they are now required to contain at least one, as one black point needs to be attributed.
Therefore, the sums in Eq. (21) start from r = 1, leading to the following result for the decomposition of K (n P ,n 1 ) ′ : It remains to study the special case of n P = 0, i.e., the case of C l = Id. This is in fact trivial. Indeed, in that case, the value of n 1 in Z j,n 1 is no longer fixed, and one must sum over all possible values of n 1 , taking into account that the case of n 1 = 1 is particular (absense of big blocks). Since K l,Id = K l l! , one obtains simply Eqs.

Decomposition of Z j over the K l,C l
To obtain the decomposition of Z j in terms of the K l,C l , one would need to invert Eqs. (22)-(23) obtained above. But we now encounter the opposite problem of that announced in the beginning of Sec. 3.3: there are too many K l,C l . Indeed, the elemen-tary quantities K l,C l are not independent, since a given cluster configuration can realise different permutations depending on the way in which the black points are attributed.
We must therefore select an independent set of K l,C l , and we make the choice of selecting the K (n P ,n 1 ) , i.e., the C l of the first type. Inverting Eq. (22) for varying n P and fixed n 1 > 1 one obtains: Since the coefficients in this sum do not depend on n 1 (provided that n 1 > 1), we can sum this relation over n 1 and write it as where we recall the notations Z j,n 1 >1 = L n 1 =2 Z j,n 1 and K (n P ,n 1 >1) = L n 1 =2 K (n P ,n 1 ) , corresponding to permutations composed of n P cycles of the same length > 1.
Consider next the case n 1 = 1. For j ≥ 2 one has simply recalling Eq. (18) and the fact that for l ≥ 2 the Z j,n 1 >1 do not appear in the decomposition of K l . However, according to Eqs. (16)-(17), the Z j,n 1 >1 do appear for l = 0 and l = 1, and one obtains Inserting the decomposition (25) of Z j,n 1 >1 into Eq. (27) one obtains the decomposition of Z 1,1 over K l and K (n P ,n 1 ) : We proceed in the same fashion for the decomposition of Z 0 ≡ Z 0,1 , finding Upon insertion of the decomposition (25) of Z j,n 1 >1 , one arrives at Since Z j = Z j,1 + Z j,n 1 >1 , we conclude from Eqs. (26)-(25) that, for any j, with the coefficients b (n P ,n 1 >1) j The decomposition of Z ≡ 0≤j≤L Z j is therefore Note that b (l) j (resp. b (n P ,n 1 >1) j ) is just the term multiplying Q j in b (l) (resp. b (n P ,n 1 >1) ).
4 Amplitudes of the eigenvalues 4.1 Decomposition of Z over the K l,D The culmination of the preceeding section was the decomposition (31) of Z j in terms of K l,C l . However, it is the K l,D which are directly related to the eigenvalues of the transfer matrix T. For that reason, we now use the relation (11) between K l,C l and K l,D to obtain the decomposition of Z j in terms of K l,D . The result is: where the coefficients b Indeed, K l = D K l,D , and since K (n P ,n 1 ) corresponds to the level l = n P n 1 , we have K (n P ,n 1 ) = D∈Sn P n 1 c(D,(n P ,n 1 )) l! K n P n 1 ,D . (Recall that (n P , n 1 ) is the class of permutations composed of n P cycles of the same length n 1 .) As explained in Sec. 3.2, the b (l,D) j are not simply equal to b l j l! because of the n 1 > 1 terms. Using Eq. (13) we find that they nevertheless obey the following relation The b (l,D) then satisfy

Relations among the K l,D
Just like the K l,C l , the K l,D are not independent, and for the same reasons. Indeed, the number of K l,D which are independent among themselves, and independent of K l ′ ,D ′ at higher levels l ′ > l, equals the number of independent K l,C l . This number in turn equals the number of integers dividing l, since the independent K l,C l are K l,Id and the K (n P ,n 1 ) with l = n P n 1 .
Therefore one can write relations between the K l,D , by selecting an independent number of K l,D and expressing the others in terms of those selected. This produces relations of the form where the K l,D l are now those not selected, and the sum of K l ′ ,D l ′ is over the D l ′ selected with l ′ ≥ l. The expressions of the coefficients e(D l , D l ′ ) depend of the choice of K l,D made. Note in particular that to obtain the e(D l , D l ′ ), the K (n P ,n 1 ) ′ must be expressed in terms of the K (n P ,n 1 ) . By combining Eqs. (23) and (24) we obtain and performing the sum over j this becomes Let us give an example of this relation: for L = 4 we find that K 3,(1,2) ′ = 2K 4,(2,2) .
The coefficients e(D l , D l ′ ) have the following properties: which can be proved using the fact that the e(D l , D l ′ ) are independent of L and that the number of eigenvalues, including degeneracies, corresponding to K l,D l is [dim(D l )] 2 n tor (L, l).
These relations between K l,D l have strong physical implications: additional degeneracies inside a level and between different levels. We shall give in the next subsection a method to determine these degeneracies, but note for now that they depend of L although the e(D l , D l ′ ) are independent of L.
We can now repeat the decompositions of the preceeding subsection, but expanding only over the selected independent K l,D . To that end, we define the coefficientsc(D, C l ) by Note that contrary to Eq. (11), the sum carries over all independent (selected) D at levels l ′ ≥ l. Because of Eq. (46), thec(D, C l ) have the following properties: if C l = Id Inserting the decomposition (47) of K l,C l into Eq. (33), we obtain the decomposition of Z over independent K l,D : where theb can be obtained using thec.
We do not have any general closed-form expression 4 forb, but in the next subsection we show how they can be determined in practice by a straightforward, though somewhat lengthy, procedure. More precisely, we determine all theb (l,D) up to l = 4, with a given convention for the choice of independent K l,D . As the b (l,D) , theb (l,D) verify except that now the sum is over independent D. This is a consequence of the properties of thec.

Method to obtain the amplitudes of the eigenvalues
Because of the additional degeneracies between the K l,D , we have not been able to find a general formula giving the total degeneracies of the eigenvalues. But, using Eq. (33) and the fact that the c(D, C l ) defined by Eq. (11) are integers, we deduce that the amplitudes of the eigenvalues are integer combinations of the b (l) l! and the b (n P ,n 1 >1) (n P n 1 )! . Determining precisely with which integers is not an easy task, and we give here a method which is operational for all values of L, though in practice it will probably become quite cumbersome for large L.
We choose K , K and K as independent K 4,D , and we express the K 4,C l in terms of those K 4,D : Next, using these expressions, we obtain K and K in terms of the independent K 4,D chosen: With these two relations we can determine the eigenvalue degeneracies between the chosen K 4,D . Recall first that according to Eq. (7) the number of eigenvalues contributing to K l,D is n tor (L, l) dim(D) and that each eigenvalue has multiplicity dim(D). Further, dim( ) = 1, dim( ) = 3 and dim( ) = 2. Consider now Eq. (60), recalling that the K l,D have been defined in Eq. (7) as a trace. We deduce that the corresponding eigenvalues must satisfy for any positive integer N. This implies that λ ,1 = λ and that λ ,2 = λ . Using this, Eq. (61) then yields for any N. This is possible provided that either λ = λ or λ = λ ,1 . But the first possibility can be excluded since, by Eqs. (52)-(53), it would imply K 4,(1,4) = 0 which is inconsistent with our hypothesis that we work on a generic lattice where all admissible K are non-zero. We conclude that λ = λ ,1 and hence that λ = λ Consider now the level l = 3. From the character table of S 3 we obtain: K 3,(1,2) ′ must be expressed terms of the independent K 4,D l chosen: We choose K and K as independent K 3,D , and we express the K 3,C l in terms of the independent K 3,D and K 4,D chosen: We obtain then the expression of K : Using again Eq. (7) and the eigenvalue identities obtained at level l = 4, this becomes from which we deduce that λ ,1 = λ , and that λ ,2 = λ ,1 . (Note that we cannot have, for example, λ = λ ,2 since these eigenvalues were shown to be independent in the preceeding analysis at level l = 4.) We can then further deduce that λ Therefore, there are 28 new eigenvalues associated to K , 28 to K , 35 to K 1 and 14 to To obtain the amplitudes associated to the eigenvalues, we use Eq. (33): and insert the expressions of the K l,C l in terms of the independent K l,D chosen above: where the amplitudes associated to the independent K l,D are: vanish, so the equations must be truncated. For example, the result (70) obtained here for L = 4, implies by truncation of the K 4,D terms that for L = 3: Likewise, the expansion (73) of Z obtained here for L = 4, implies by truncation that for L = 3: From these equations, it is then simple to obtain the amplitudes of eigenvalues for L = 3.

Particular non-generic lattices
The degeneracies we have obtained apply to the case of a generic lattice. In the case of a specific lattice, i.e., one having extra non-generic symmetries, there might be additional degeneracies. An example is the case of a square or a honeycomb lattice, because of the invariance of the lattice under reflection by its symmetry axis.
Specifically, the transfer matrix T L of the square lattice commutes with the dihedral group D L , since the lattice enjoys both translational and reflectional symmetries in the space perpendicular to the transfer direction. Likewise, for the honeycomb lattice, the symmetry group is D L

2
. Those groups act on the right time slice, not on the left one (i.e., the black points), and thus commute with the symmetry group S l at level l of the bridges. There are therefore additional degeneracies inside a given K l,D .
In the case of the dihedral group, since all its irreps are of dimension 1 or 2, there are additional degeneracies between pairs of eigenvalues, as observed in Ref. [3]. The method to determine precisely these degeneracies is to decompose the space at level l of symmetry D into irreps of the dihedral group, and to count the number of irreps of dimension 2.
Furthermore, in the case of a square or a hexagonal lattice, there are yet additional degeneracies (which do not exist for a generic lattice with a dihedral symmetry). Indeed, for these two lattices, the Z L,n 1 with n 1 > 1 are found to be zero: at level l = L no permutation between bridges is allowed. At this level there is thus only one eigenvalue of total degeneracy D∈S l [dim(D)] 2b (L,D) = b (L) . Finally, in the case of a square lattice, there appears to be additional "accidental" degeneracies: for example, an eigenvalue at level 1 coincides with an eigenvalue at level 2, as observed in Ref. [3].

Conclusion
To summarise, we have generalised the combinatorial approach developed in Ref. [2] for cyclic boundary conditions to the case of toroidal boundary conditions. In particular, we have obtained the decomposition of the partition function for the Potts model on finite tori in terms of the generalised characters K l,D . This decomposition is considerably more difficult to interpret than in the cyclic case, as some eigenvalues coincide between different levels l for all values of Q. We have nevertheless succeeded in giving an operational method of determining the amplitudes of the eigenvalues as well as their generic degeneracies.
The eigenvalue amplitudes are instrumental in determining the physics of the Potts model, in particular in the antiferromagnetic regime [8]. Generically, this regime belongs to a so-called Berker-Kadanoff (BK) phase in which the temperature variable is irrelevant in the renormalisation group sense, and whose properties can be obtained by analytic continuation of the well-known ferromagnetic phase transition [9].