david e. evans and mathew pugh- ocneanu cells and boltzmann weights for the su(3) ade graphs

8/3/2019 David E. Evans and Mathew Pugh- Ocneanu cells and Boltzmann weights for the SU(3) ADE graphs

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Munster J. of Math. 2 (2009), 95142 Munster Journal of Mathematics

urn:nbn:de:hbz:6-10569512950 c Munster J. of Math. 2009

Ocneanu cells and Boltzmann weights for the

SU(3) ADE graphsDavid E. Evans and Mathew Pugh

(Communicated by Siegfried Echterhoff)

Dedicated to Joachim Cuntz on the occasion of his 60th birthday

Abstract. We determine the cells, whose existence has been announced by Ocneanu, on

all the candidate nimrep graphs except E(12)4 proposed by di Francesco and Zuber for the

SU(3) modular invariants classified by Gannon. This enables the Boltzmann weights to becomputed for the corresponding integrable statistical mechanical models and provide theframework for studying corresponding braided subfactors to realize all the SU(3) modularinvariants as well as a framework for a new SU(3) planar algebra theory.

1. Introduction

In the last twenty years, a very fruitful circle of ideas has developed link-ing the theory of subfactors with modular invariants in conformal field theory.Subfactors have been studied through their paragroups, planar algebras andhave serious contact with free probability theory. The understanding and clas-sification of modular invariants is significant for conformal field theory andtheir underlying statistical mechanical models. These areas are linked through

the use of braided subfactors and -induction which in particular for SU(2)subfactors and SU(2) modular invariants invokes ADE classifications on bothsides. This paper is the first of our series to study more precisely these con-nections in the context of SU(3) subfactors and SU(3) modular invariants.The aim is to understand them not only through braided subfactors and -induction but introduce and develop a pertinent planar algebra theory and freeprobability.

A group acting on a factor can be recovered from the inclusion of its fixedpoint algebra. A general subfactor encodes a more sophisticated symmetry ora way of handling non group like symmetries including but going beyond quan-tum groups [18]. The classification of subfactors was initiated by Jones [29]


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96 David E. Evans and Mathew Pugh

who found that the minimal symmetry to understand the inclusion is throughthe Temperley-Lieb algebra. This arises from the representation theory ofSU(2) or dually certain representations of Hecke algebras. All SU(2) modularinvariant partition functions were classified by Cappelli, Itzykson and Zuber[11, 12] using ADE Coxeter-Dynkin diagrams and their realization by braided

subfactors is reviewed and referenced in [15]. There are a number of invari-ants (encoding the symmetry) one can assign to a subfactor, and under certaincircumstances they are complete at least for hyperfinite subfactors. Popa [39]axiomatized the inclusions of relative commutants in the Jones tower, andJones [30] showed that this was equivalent to his planar algebra description.Here one is naturally forced to work with nonamenable factors through freeprobabilistic constructions e.g. [27]. In another vein, Banica and Bisch [3]understood the principal graphs, which encode only the multiplicities in theinclusions of the relative commutants, and more generally nimrep graphs interms of spectral measures, and so provide another way of understanding the

subfactor invariants.In our series of papers we will look at this in the context of SU(3), throughthe subfactor theory and their modular invariants, beginning here and contin-uing in [20, 21, 22, 23, 24]. The SU(3) modular invariants were classified byGannon [26]. Ocneanu [37] announced that all these modular invariants wererealized by subfactors, and most of these are understood in the literature andwill be reviewed in the sequel [20]. A braided subfactor automatically givesa modular invariant through -induction. This -induction yields a represen-tation of the Verlinde algebra or a nimrep - which yields multiplicity graphsassociated to the modular invariants (or at least associated to the inclusion,

as a modular invariant may be represented by wildly differing inclusions andso may possess inequivalent but isospectral nimreps, as is the case for E(12)).In the case of the SU(3) modular invariants, candidates of these graphs wereproposed by di Francesco and Zuber [14] by looking for graphs whose spectrumreproduced the diagonal part of the modular invariant, aided to some degreeby first listing the graphs and spectra of fusion graphs of the finite subgroupsof SU(3). In the SU(2) situation there is a precise relation between the ADECoxeter-Dynkin graphs and finite subgroups of SU(2) as part of the McKaycorrespondence. However, for SU(3), the relation between nimrep graphs andfinite subgroups ofSU(3) is imprecise and not a perfect match. For SU(2), an

affine Dynkin diagram describing the McKay graph of a finite subgroup givesrise to a Dynkin diagram describing a nimrep or the diagonal part of a modularinvariant by removing the vertex corresponding to the identity representation.Di Francesco and Zuber found graphs whose spectrum described the diagonalpart of a modular invariant by taking the list of McKay graphs of finite sub-groups of SU(3) and removing vertices. Not every modular invariant couldbe described in this way, and not every finite subgroup yielded a nimrep for amodular invariant. In higher rank SU(N), the number of finite subgroups willincrease but the number of exceptional modular invariants should decrease, so

Munster Journal of Mathematics Vol. 2 (2009), 95142


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Ocneanu cells and Boltzmann weights for SU(3) ADE graphs 97

this procedure is even less likely to be accurate. Evans and Gannon have sug-gested an alternative way of associating finite subgroups to modular invariants,by considering the largest finite stabilizer groups [16].

A modular invariant which is realized by a subfactor will yield a graph. Toconstruct these subfactors we will need some input graphs which will actu-

ally coincide with the output nimrep graphs - SU(3) ADE graphs. The aimof this series of papers is to study the SU(3) ADE graphs, which appear inthe classification of modular invariant partition functions from numerous view-points including the determination of their Boltzmann weights in this paper,representations of SU(3)-Temperley-Lieb or Hecke algebra [20], a new notionof SU(3)-planar algebras [21] and their modules [22], and spectral measures[23, 24].

As pointed out to us by Jean-Bernard Zuber, there is a renewal of interest(by physicists) in these SU(3) and related theories, in connection with topo-logical quantum computing [1] and by Joost Slingerland in connection with

condensed matter physics [2] where we see that -induction is playing a keyrole.We begin however in this paper by computing the numerical values of the

Ocneanu cells, and consequently representations of the Hecke algebra, for theADE graphs. These cells give numerical weight to Kuperbergs [32] diagramof trivalent verticescorresponding to the fact that the trivial representationis contained in the triple product of the fundamental representation of SU(3)through the determinant. They will yield in a natural way, representations ofan SU(3)-Temperley-Lieb or Hecke algebra. (For SU(2) or bipartite graphs,the corresponding weights (associated to the diagrams of cups or caps), arise

in a more straightforward fashion from a Perron-Frobenius eigenvector, givinga natural representation of the Temperley-Lieb algebra or Hecke algebra). We

have been unable thus far to compute the cells for the exceptional graph E(12)4 .This graph is meant to be the nimrep for the modular invariant conjugate to

the Moore-Seiberg invariant E(12)MS [33]. However we will still be able to realizethis modular invariant by subfactors in [20] using [19]. For the orbifold graphs

D(3k), k = 2, 3, . . . , orbifold conjugate D(n), n = 6, 7, . . . , and E(12)1 wecompute solutions which satisfy some additional condition, but for the othergraphs we compute all the Ocneanu cells, up to equivalence. The existenceof these cells has been announced by Ocneanu (e.g. [36, 37]), although the

numerical values have remained unpublished. Some of the representations ofthe Hecke algebra have appeared in the literature and we compare our results.

For the A graphs, our solution for the Ocneanu cells W gives an identicalrepresentation of the Hecke algebra to that of Jimbo et al. [28] given in (21).Our cells for the A(n) graphs give equivalent Boltzmann weights to thosegiven by Behrend and Evans in [4]. In [14], di Francesco and Zuber give arepresentation of the Hecke algebra for the graphs D(6) and E(8), whilst in[41] a representation of the Hecke algebra is computed for the graphs E(12)1 andE(24). Our solutions for the cells W give an identical Hecke representation for



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E(8) and an equivalent Hecke representation for E(12)1 . However, for E(24), ourcells give inequivalent Boltzmann weights. In [25], Fendley gives Boltzmannweights for D(6) which are not equivalent to those we obtain, but which areequivalent if we take one of the weights in [25] to be the complex conjugate ofwhat is given.

Subsequently, we will use these weights, their existence and occasionallymore precise use of their numerical values. Here we outline some of the flavorof these applications. We use these cells to define an SU(3) analogue of theGoodman-de la Harpe Jones construction of a subfactor, where we embedthe SU(3)-Temperley-Lieb or Hecke algebra in an AF path algebra of theSU(3) ADEgraphs. Using this construction, we realize all the SU(3) modularinvariants by subfactors [20].

We will then [21, 22] look at the SU(3)-Temperley-Lieb algebra and theSU(3)-GHJ subfactors from the viewpoint of planar algebras. We give a di-agrammatic representation of the SU(3)-Temperley-Lieb algebra, and show

that it is isomorphic to Wenzls representation of a Hecke algebra. Generaliz-ing Joness notion of a planar algebra, we construct an SU(3)-planar algebrawhich will capture the structure contained in the SU(3) ADEsubfactors. Weshow that the subfactor for an ADE graph with a flat connection has a de-scription as a flat SU(3)-planar algebra. We introduce the notion of modulesover an SU(3)-planar algebra, and describe certain irreducible Hilbert SU(3)-T L-modules. A partial decomposition of the SU(3)-planar algebras for theADEgraphs is achieved. Moreover, in [23, 24] we consider spectral measuresfor the ADE graphs in terms of probability measures on the circle T. We gen-eralize this to SU(3), and in particular obtain spectral measures for the SU(3)

graphs. We also compare various Hilbert series of dimensions associated toADE models for SU(2), and compute the Hilbert series of certain q-deformedCalabi-Yau algebras of dimension 3.

In Section 2, we specify the graphs we are interested in, and in Section3 recall the notion of cells due to Ocneanu which we will then compute inSections 4 - 14.

2. ADE GraphsWe enumerate the graphs we are interested in. These will eventually provide

the nimrep classification graphs for the list ofSU(3) modular invariants, but atthis point, they will only provide a framework for some statistical mechanicalmodels with configurations spaces built from these graphs together with someBoltzmann weights which we will need to construct. However, for the sake ofclarity of notation, we start by listing the SU(3) modular invariants. Thereare four infinite series of SU(3) modular invariants: the diagonal invariants,labelled by A, the orbifold invariants D, the conjugate invariants A, and theorbifold conjugate invariants D. These will provide four infinite families ofgraphs, written as A, the orbifold graphs D, the conjugate graphs A, and theorbifold conjugate graphs D, shown in Figures 4, 7, 10, 11 and 12. There



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are also exceptional SU(3) modular invariants, i.e. invariants which are notdiagonal, orbifold, or their conjugates, and there are only finitely many of

these. These are E(8) and its conjugate, E(12), E(12)MS and its conjugate, andE(24). The exceptional invariants E(12) and E(24) are self-conjugate.

The modular invariants arising from SU(3)k conformal embeddings are:

D(6): SU(3)3 SO(8)1, also realized as an orbifold SU(3)3/Z3, E(8): SU(3)5 SU(6)1, plus its conjugate, E(12): SU(3)9 (E6)1, E(12)MS : Moore-Seiberg invariant, an automorphism of the orbifold in-

variant D(12) = SU(3)9/Z3, plus its conjugate, E(24): SU(3)21 (E7)1.

These modular invariants will be associated with graphs, as follows. Therewill be one graph E(8) for the E(8) modular invariant and its orbifold graphE(8) for its conjugate invariant as in Figure 13. The modular invariants E(12)MSand its conjugate will be associated to the graphs E(12)5 and E(12)4 respectivelyas in Figure 15. The exceptional invariant E(12) is self-conjugate but has as-sociated to it two isospectral graphs E(12)1 and E(12)2 as in Figure 14. Theinvariant E(24) is also self-conjugate and has associated to it one graph E(24)as in Figure 16. The modular invariants themselves play no role in this pa-per other than to help label these graphs. In the sequel to this paper [20] wewill use the Boltzmann weights obtained here to construct braided subfactors,which via -induction [5, 6, 7, 8, 9, 10] will realize the corresponding modularinvariants. Furthermore, -induction naturally provides a nimrep or repre-sentation of the original fusion rules or Verlinde algebra. The corresponding

nimreps will then be computed and we will recover the original input graph.The theory of -induction will guarantee that the spectra of these graphs aredescribed by the diagonal part of the corresponding modular invariant. Thusdetailed information about the spectra of these graphs will naturally followfrom this procedure and does not need to be computed at this stage. Many ofthese modular invariants are already realized in the literature and this will bereviewed in the sequel to this paper [20].

3. Ocneanu Cells

Let be SU(3) and its irreducible representations. One can associateto a McKay graph G whose vertices are labelled by the irreducible repre-sentations of , where for any pair of vertices i, j the number of edgesfrom i to j are given by the multiplicity of j in the decomposition of i into irreducible representations, where is the fundamental irreducible rep-resentation of SU(3), and which, along with its conjugate representation ,

generates . The graph G is made of triangles, corresponding to the factthat the fundamental representation satisfies 1. We define mapss, r from the edges of G to its vertices, where for an edge , s() denotesthe source vertex of and r() its range vertex. For the graph G, a triangle



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()ijk = i- j

- k

- i is a closed path of length 3 consisting of edges

, , ofG such that s() = r() = i, s() = r() = j and s() = r() = k.For each triangle ()ijk , the maps , and are composed:

i

iddet-

i id-

k id-

j id-

i,

and since i is irreducible, the composition is a scalar. Then for every suchtriangle on G there is a complex number, called an Ocneanu cell. There isa gauge freedom on the cells, which comes from a unitary change of basis inHom[i , j] for every pair i, j.

These cells are axiomatized in the context of an arbitrary graph G whoseadjacency matrix has Perron-Frobenius eigenvalue [3] = [3]q, although in prac-tice it will be one of the ADE graphs. Note however we do not require Gto be three-colorable (e.g. the graphs A which will be associated to theconjugate modular invariant). Here the quantum number [m]

qbe defined as

[m]q = (qm qm)/(q q1). We will frequently denote the quantum number[m]q simply by [m], for m N. Now [3]q = q2 + 1 + q2, so that q is easilydetermined from the eigenvalue ofG. The quantum number [2] = [2]q is thensimply q + q1. If G is an ADE graph, the Coxeter number n of G is thenumber in parentheses in the notation for the graph G, e.g. the exceptionalgraph E(8) has Coxeter number 8, and q = ei/n. With this q, the quantumnumbers [m]q satisfy the fusion rules for the irreducible representations of thequantum group SU(2)n, i.e.

(1) [a]q [b]q = c [c]q,where the summation is over all integers |b a| c min(a + b, 2n a b)such that a + b + c is even.

We define a type I frame in an arbitrary G to be a pair of edges , whichhave the same start and endpoint. A type II frame will be given by four edgesi, i = 1, 2, 3, 4, such that s(1) = s(4), s(2) = s(3), r(1) = r(2) andr(3) = r(4).

Definition 3.1 ([37]). Let

Gbe an arbitrary graph with Perron-Frobenius

eigenvalue [3] and Perron-Frobenius eigenvector (i). A cell system W onG is a map that associates to each oriented triangle ()ijk in G a complexnumber W

()ijk

with the following properties:

(i) for any type I frame in G we have

(2)



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(ii) for any type II frame in G we have

(3)

In [32], Kuperberg defined the notion of a spidera way of depicting theoperations of the representation theory of groups and other group-like objectswith certain planar graphs, called webs (hence the term spider). Certainspiders were defined in terms of generators and relations, isomorphic to therepresentation theories of rank two Lie algebras and the quantum deformationsof these representation theories. This formulation generalized a well-known

construction for A1 = su(2) by Kauffman [31]. For the A2 = su(3) case, theA2 webs are illustrated in Figure 1.

Figure 1. A2 webs

The A2 web space generated by these A2 webs satisfy the Kuperberg rela-

tions, which are relations on local parts of the diagrams:

K1:

K2:

K3:

The rules (2), (3) correspond precisely to evaluating the Kuperberg relationsK2, K3 respectively, associating a cell W(,,) to an incoming trivalentvertex, and W(,,) to an outgoing trivalent vertex, as in Figure 2.

Figure 2. Cells associated to trivalent vertices



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We define the connection

X1,23,4 =

l1- i

k

3 ? 4- j

2?

for G by(4) X1,23,4 = q

23 1,32,4 q

13U1,23,4 ,

where U1,23,4 is given by the representation of the Hecke algebra, and is definedby

(5) U1,23,4 =

1s(1)1r(2)

W((,3,4)j,l,k )W((,1,2)j,l,i ).

This definition of the connection is really Kuperbergs braiding of [32].

The above connection corresponds to the natural braid generator gi, whichis the Boltzmann weight at criticality, and which satisfy

gigj = gj gi if |j i| > 1,(6)gigi+1gi = gi+1gigi+1.(7)

It was claimed in [36] that the connection satisfies the unitarity property ofconnections

(8)

3,4X1,23,4 X

1,

23,4 = 1,12,2 ,

and the Yang-Baxter equation

(9)

1,2,3

X1,21,2 X3,41,3 X

3,52,6 =

1,2,3

X3,21,1 X1,32,6 X

4,52,3 .

The Yang-Baxter equation (9) is represented graphically in Figure 3. We givea proof that the connection (4) satisfies these two properties.

Figure 3. The Yang-Baxter equation

Lemma 3.2. If the conditions in Definition 3.1 are satisfied, the connectiondefined in (4) satisfies the unitarity property (8) and the Yang-Baxter equation(9).



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Proof. We first show unitarity.3,4

X1,23,4 X1,

23,4

= 3,4

q 23 1,32,4 q 13

1

s(1)r(2) W3,4,W1,2,

q23 1,32,4 q

13

1

s(1)r(2)W1 ,2,W3,4,

= 1,13,3 +3,4,

1

2s(1)2r(2)

W3,4,W1,2,W1,2,W3,4,

3,4,1

s(1)r(2) q1,32,4W1,2,W3,4,

+q11,32,4W3,4,W1,2,

= 1,13,3 +,

1

2s(1)2r(2)

W1,2,W1,2,[2]s(3)r(4),

(q + q1)

1

s(1)r(2)W1,2,W1,2,

= 1,13,3 ,

since q + q1 = [2], where we have used Ocneanus type I equation (3) in the

penultimate equality.We now show that the connection satisfies the Yang-Baxter equation. For

the left hand side of (9) we have1,2,3

X1,21,2 X3,41,3X

3,52,6

=

1,2,3

q23 1,12,2 q

13U1,21,1

q

23 1,33,4 q

13U3,41,3

q23 2,36,5 q

13U3,52,6 = q21,32,45,6 q1,3 U4,52,6 q5,6 U3,41,2 q5,6 U3,41,2

+

3

U3,41,3 U3,52,6 +

2

U3,21,2 U4,52,6 + 5,6

1,2

U1,21,2 U3,41,2

q1

i

U1,21,2 U3,41,3 U3,52,6

= q21,32,45,6 q1,3 U4,52,6 2q5,6 U3,41,2+

3

U3,41,3 U3,52,6 +

2

U3,21,2 U4,52,6



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+ 5,61,2,

1

s(1)r(2)s(3)r(4)W1,2,W1,2,W1,2,W3,4,

q1

i,,1

2s(1)r(2)r(4)s(2)r(6)W1,2,W3,4,

W1,2,W1,3,W2,6,W1,5,= q21,32,45,6 q1,3 U4,52,6 2q5,6 U3,41,2

+

3

U3,41,3 U3,52,6 +

2

U3,21,2 U4,52,6

+ 5,6,

1

s(1)r(2)s(3)r(4)W1,2,W3,4, [2]r(2)s(1),

q1,

1

2

s(1)r(2)r(4)r(6)

W1,2,W3,4,

,6,5r(2)r(6)r(4) + ,5,6s(1)r(2)r(6)= q21,32,45,6 q1,3 U4,52,6 q5,6 U3,41,2 +

3

U3,41,3 U3,52,6

+

2

U3,21,2 U4,52,6 q11

s(1)W1,2,6W3,4,5 .

Computing the right hand side of (9) in the same way, we arrive at the sameexpression.

4. Computation of the cells W for ADE graphsIn the remaining sections we will compute cells systems W for each ADE

graph G, with the exception of the graph E(12)4 .Let (,,)i,j,k be the triangle i

- j

- k

- i in G. For most of the

ADE graph, using the equations (2) and (3) only, we can compute the cellsup to choice of phase W((,,)i,j,k ) = ,,i,j,k |W((,,)i,j,k )| for some ,,i,j,k T,and also obtain some restrictions on the values which the phases ,,i,j,k may

take. However, for the graph D(n)

, n = 5, 6, . . . , we impose aZ3 symmetryon our solutions, whilst for the graphs D(3k), k = 2, 3, . . . , and E(12)1 we seek

an orbifold solution obtained using the identification of the graphs D(3k), E(12)1as Z3 orbifolds ofA(3k), E(12)2 respectively. There is still much freedom in theactual choice of phases, so that the cell system is not unique. We thereforedefine an equivalence relation between two cell systems:

Definition 4.1. Two families of cells W1, W2 which give a cell system forG are equivalent if, for each pair of adjacent vertices i, j of G, we can find afamily of unitary matrices (u(1, 2))1,2 , where 1, 2 are any pair of edges



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from i to j, such that

(10) W1((,,)i,j,k ) =

,,

u(, )u(, )u(, )W2((,,)

i,j,k ),

where the sum is over all edges from i to j, from j to k, and from k to

i.Lemma 4.2. Let W1, W2 be two equivalent families of cells, and X

(1), X(2)

the corresponding connections defined using cells W1, W2 respectively. ThenX(1) and X(2) are equivalent in the sense of [18, p.542], i.e. there exists a setof unitary matrices (u(, )), such that

X(1)1,23,4 =

i

u(3, 3)u(4, 4)u(1, 1)u(2, 2)X(2)1,23,4 .

Let Wl((,,)i,j,k ) = (l),,i,j,k |Wl((,,)i,j,k )|, for l = 1, 2, be two families of

cells which give cell systems. If |W1((,,)

i,j,k )| = |W2((,,)

i,j,k )|, so that W1and W2 differ only up to phase choice, then the equation (10) becomes(11)

(1),,i,j,k =

,,

u(, )u(, )u(, )(2),,i,j,k .

For graphs with no multiple edges we write i,j,k for the triangle (,,)i,j,k .For such graphs, two solutions W1 and W2 differ only up to phase choice, and(11) becomes

(12) (1)i,j,k = uuu

(2)i,j,k,

where u, u, u T

and is the edge from i to j, the edge from j to k and the edge from k to i.We will write U(x,y) for the matrix indexed by the vertices ofG, with entries

given by U1,23,4 for all edges i, i = 1, 2, 3, 4 on G such that s(1) = s(3) = x,r(2) = r(4) = y, i.e. [U(

s(1),r(2))]r(1),r(3) = U1,23,4 .We first present some relations that the quantum numbers [a]q satisfy, which

are easily checked:

Lemma 4.3.

(i) If q = exp(i/n) then [a]q = [n a]q, for any a = 1, 2, . . . , n 1,(ii) For any q, [a]q

[a

2]q = [2a

2]q/[a

1]q, for any a

N,

(iii) For anyq, [a]2q [a1]q[a+1]q = 1 and[a]q[a+b]q [a1]q[a+b+1]q =[b + 1]q, for any a N.

5. A graphsThe infinite graph A() is illustrated in Figure 4, whilst for finite n, the

graphs A(n) are the subgraphs ofA(), given by all the vertices (1, 2) suchthat 1 + 2 n 3, and all the edges in A() which connect these ver-tices. The apex vertex (0, 0) is the distinguished vertex. For the triangle(i1,j1)(i2,j2)(i3,j3) = (i1, j1) - (i2, j2) - (i3, j3) - (i1, j1) in A(n) we



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Figure 4. The infinite graph A()

will use the notation W(i,j) for the cell W((i,j)(i+1,j)(i,j+1)) and W(i,j) forthe cell W((i+1,j)(i,j+1)(i+1,j+1)).Theorem 5.1. There is up to equivalence a unique set of cells for A(n), n 0, if n = 2m,

x =

[2][2i]+[4i]

[2i1][2i][2i+1] for i = 1, . . . , m 1,[2]

[2m1]2 for i = m,

[2][2i][4n4i][2i1][2i][2i+1] for i = m + 1, . . . , 2m 1,

,

and ifn = 2m + 1,

x = [2][2i]+[4i]

[2i1][2i][2i+1] for i = 1, . . . , m ,[2][2i][4n4i][2i1][2i][2i+1] for i = m + 1, . . . , 2m,

.

Lemma 7.5. The weights in the representation of the Hecke algebra givenabove for A(2n) are equivalent to the Boltzmann weights at criticality given byBehrend-Evans in [4].

Proof. To make our notation the same as that of [4] one replaces i with a/2. Tosee that the absolute values of our weights are equal to those of the Boltzmannweights in [4] one needs the following relations on the quantum numbers:

[2i] + [1] = [2i + 1]q

[4i + 2]q

[2i 1]q , [2i] [1] = [2i 1]q

[4i + 2]q

[2i + 1]q,

where q =

q (q = ei/n). Again, a bit more work is required for [U(i,i)]i,i.

For equivalence we make the same choice of (ui,j)i,j as for A(2n+1).

8. D graphsThe graphs D(n) are illustrated in Figure 12. We label its vertices by il, jl

and kl, l = 1, . . . , (n1)/2, which we have illustrated in Figure 12 for n = 9.We consider first the graphs D(2n+1). The Perron-Frobenius weights are

il = jl = kl = [2l 1], l = 1, . . . , n. Since the graph has a Z3 symmetry,we will seek Z3-symmetric solutions (up to choice of phase), i.e. |Wip,jq,kr |2 =|Wiq ,jr,kp |2 = |Wir,jp,kq |2 =: |Wp,q,r |2, p,q,r {1, . . . , n}. Using this notation,we have the following equations from type I frames:

|W1,2,2|2 = [2][3],(30)|Wl,l,l+1|2 + |Wl,l+1,l+1|2 = [2][2l 1][2l + 1], l = 2, . . . , n 1,(31)|Wl1,l,l|2 + |Wl,l,l|2 + |Wl,l,l+1|2 = [2][2l 1]2, l = 2, . . . , n 1,(32)

|Wn1,n,n|2 + |Wn,n,n|2 = [2]3,(33)



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Figure 12. D(n) for n = 6, 7, 8, 9

and from type II frames we have:

(34) |Wl1,l,l|2|Wl,l,l+1|2 = [2l 3][2l 1]2[2l + 1],for l = 2, . . . , n 1, and(35) |Wl1,l,l|2( 1

[2l 3] |Wl1,l1,l|2 +

1

[2l 1] |Wl,l,l|2) = [2l 3][2l 1]2,

for l = 2, . . . , n, which are exactly those for the type I and type II frames forthe graph A(2n+1). Since the Perron-Frobenius weights and Coxeter numberare also the same as for A(2n+1), the cells |Wp,q,r,| follow.

From the type II frame consisting of the vertices il, jl, il+1 and jl+1 we havethe following restriction on the choice of phase

il,jl,kl+1il,jl+1,klil+1,jl,klil+1,jl+1,kl+1(36)

= il,jl,klil,jl+1,kl+1il+1,jl,kl+1il+1,jl+1,kl .Theorem 8.1. EveryZ3-symmetric solution for the cells W ofD(2n+1), n

david e. evans and mathew pugh- ocneanu cells and boltzmann weights for the su(3) ade graphs

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