Two elliptic height models with factorized domain wall partition functions
aa r X i v : . [ m a t h - ph ] N ov TWO ELLIPTIC HEIGHT MODELS WITH FACTORIZEDDOMAIN WALL PARTITION FUNCTIONS
O FODA, M WHEELER AND M ZUPARIC
Abstract.
We obtain factorized domain wall partition functions in two el-liptic height models: A Felderhof-type model, which is new, and APerk-Schultz-type gl (1 |
1) model of Deguchi and Martin. Introduction
Factorization in trigonometric vertex models.
In [1], we obtained fac-torized domain wall partition functions (DWPF’s) in two series of trigonometricvertex models: The N -state Deguchi-Akutsu models, for N ∈ { , , } (andconjectured the result for N ≥ The gl ( r + 1 | s + 1) Perk-Schultz models, { r, s } ∈ N (where given the symmetries of these models, the result is independentof r and s ).0.2. Asymmetry.
These models were characterized by an asymmetry of the vertexweights under conjugation of state variables. For example, in the Deguchi-Akutsumodel, with state variables σ ∈ { , · · · , N } , the vertex weights are non-invariantunder the conjugation σ → ( N − σ + 1). In the Perk-Schultz models, a similarproperty holds. Since one can trace the factorization of the DWPF’s obtainedin [1] to this asymmetry, it is natural to look for height models with the sameproperty.0.3. Factorization in elliptic height models.
In this work we consider heightmodels, where the state variables are heights, that live on the corners of the faces of asquare lattice. A weight is assigned to each face. The weights are elliptic functionsof the corresponding rapidities, external fields (if any) and height variables. Asin [1], the models in this work are characterized by an asymmetry of the weights, inthe sense that the weights of certain vertices (called line-permuting vertices) havedifferent zeros, which leads to the factorization of the DWPF’s.0.4.
Summary of results.
We obtain factorized DWPF’s for two elliptic heightmodels: A Felderhof-type model, which (to the best of our knowledge) is new,and A Perk-Schultz-type gl (1 |
1) model of Deguchi and Martin [3]. These arethe first examples of DWPF’s for elliptic and/or height models.0.5.
Outline of paper.
In Section , we collect a number of basic definitionsrelated to elliptic height models to make the paper reasonably self-contained. InSection , we introduce a new Felderhof-type elliptic height model and obtainthe corresponding factorized DWPF. In Section , we do the same for the gl (1 | contains brief remarks.The presentation is elementary in the hope that the paper will be reasonably self-contained. Mathematics Subject Classification.
Primary 82B20, 82B23.
Key words and phrases.
Elliptic height models. Domain wall boundary conditions. For a review of previous results on the subject, see [2]. Also known as face or interaction-round-face (IRF) models. Height models
Faces and corners.
We work on a square lattice, as in Figure , with L square faces f ij , where 1 ≤ i ≤ L increases from top to bottom, and 1 ≤ j ≤ L increases from left to right. f ij has four corners that are labelled from top-leftclockwise as { c i,j , c i,j +1 , c i +1 ,j +1 , c i +1 ,j } . f f f L f L f L f LL h h h L h L h LL h u , p u , p u L , p L v ,q v ,q v L ,q L Figure 1.
A square lattice with L faces f ij . Rapidities { u, v } and external fields { p, q } flow along lines that cross the faces. Height variables h ij live on the corners. Heights and restrictions.
We assign each corner c ij a height variable h ij ,0 ≤ i, j ≤ L . In height models such as Baxter’s solid-on-solid model [4], the heightsare integral (possibly up to an overall shift). In the model of Section , the heightvariables depend linearly on the external fields which are continuous parameters,so they are no longer integral. We define the heights and the restrictions that theyobey on a model by model basis in Sections and .1.3. Flow lines, orientations and variables.
There are L horizontal and L vertical lines that intersect at the middle points of f ij . They indicate the flow ofrapidities and external fields through f ij . We assign the i -th horizontal line anorientation from left to right, a complex rapidity u i and a complex external field p i . We assign the j -th vertical line an orientation from bottom to top, a complexrapidity v j and a complex external field q j , as in Figure .1.4. Weights and Yang-Baxter equations.
We assign each f ij a weight w ij that depends on the height variables on its corners, the difference of the rapidityvariables and the two external field variables (if any) flowing through it. The weightssatisfy a set of Yang-Baxter equations. The weights and Yang-Baxter equations ofthe models discussed in this paper are given in Sections and . LLIPTIC HEIGHT MODELS AND FACTORIZED DWPF 3
Elliptic functions and a theorem.
Following the conventions used in [4],Chapter , we consider the elliptic function H ( u ) = 2 q / sin πu I ∞ Y n =1 − q n cos πuI + q n − q n (1)where u ∈ C , q = exp − πI ′ I , 2 I and 2 I ′ (usually called 2 K and 2 K ′ ) arerespectively the (real) width and height of an (upright) rectangle R in the complex u -plane, so that 0 < q <
1. It is convenient to define[ u ] = H ( u )2 q / (2)which is entire and satisfies the quasi-periodicity properties[ u + 2 I ] = − [ u ] (3)[ u + 2 iI ′ ] = − q exp − πiuI [ u ] (4) Theorem 1. If f ( u ) is an entire function that satisfies the quasi-periodicity con-ditions f ( u + 2 I ) = ( − ) L f ( u ) (5) f ( u + 2 iI ′ ) = − q L exp − πi ( Lu − η ) I f ( u ) (6)then f ( u ) = κ L − Y j =1 [ u − ζ j ] [ u − η + L − X j =1 ζ j ] (7)where κ and ζ , . . . , ζ L − are constants. Proof.
This is a refinement of Theorem of [4], and the proof uses a similarargument. Choose the period rectangle R such that f ( u ) has no zeros on theboundary ∂R , and integrate f ′ ( u ) f ( u ) on the anti-clockwise contour ∂R . From thequasi-periodicity conditions it follows that I ∂R f ′ ( u ) f ( u ) du = 2 πiL (8)Hence the sum of residues of f ′ ( u ) f ( u ) in R is equal to L , showing f ( u ) has exactly L zeros in R (counting a zero of order n with multiplicity n ). Writing the L zeros as ζ , . . . , ζ L , we define the function φ ( u ) = Q Lj =1 [ u − ζ j ]. Since ddu log( f ( u ) /φ ( u )) isdoubly periodic (by construction) and holomorphic (also by construction) one has ddu log f ( u ) φ ( u ) = λ (9) O FODA, M WHEELER AND M ZUPARIC where λ is a constant. Integrating, we obtain f ( u ) = κe λu Q Lj =1 [ u − ζ j ]. Usingthe quasi-periodicity conditions of f ( u ), we can, without loss of generality, choose λ = 0 and ζ L = η − P L − j =1 ζ j , which concludes the proof.2. A Felderhof-type height model
In this section, we introduce an elliptic height model with weights that dependon rapidities, external fields and height variables. In the trigonometric limit, itreduces to the first in a series of models introduced by Deguchi and Akutsu in [5].In that same limit, and decoupling the dependence on the heights , it reduces tothe trigonometric limit of the elliptic Felderhof vertex model, which is the 2-stateDeguchi-Akutsu model [6].2.1. Notation.
Given the rapidities { u, v } ∈ C , external fields { p, q } ∈ C , anupper-left corner height h ∈ C , { ∆ , ∆ } ∈ { , } , ∆ ∈ { , , } , we use thenotation W uv h h + q − ∆1 h + p − ∆2 h + q + p − ∆3 (10)for the weight assigned to the vertex represented in Figure . u , p v , qhh + p − ∆2 h + q − ∆1 h + q + p − ∆3 Figure 2.
A Felderhof-type face configuration.
Height restrictions.
For p = q = , we require that the heights satisfy thesame restriction as in Baxter’s solid-on-solid model [4], up to a normalization. Moreprecisely, h i,j − h i +1 ,j = ± , h i,j − h i,j +1 = ±
12 (11)2.3.
The crossing parameter = . The vertex weights will be parametrized interms of the elliptic functions [ u ]. As defined in Equations and , [ u ] depends onthe real parameters, I and I ′ , which are the magnitudes of the half-periods of [ u ].In the Felderhof-type model discussed in this section, we set I = 1 . This can be achieved, for example, by introducing a parameter ξ ∈ i R , shifting all heightvariables by ξ (the Yang-Baxter equations remain satisfied), then taking the limit ξ → i ∞ . In the sequel, we simply say ‘vertex’ instead of ‘face configuration’. In the limit of zero external fields, that is p = q = , this is equivalent to setting the crossingparameter in Baxter’s solid-on-solid model to the free fermion point. For details, see [4] LLIPTIC HEIGHT MODELS AND FACTORIZED DWPF 5
The weights.
In the above notation, the non-zero weights are W uv h h + qh + p h + q + p = a + ( u, v, p, q ) = [ u − v + p + q ] (12) W uv h h + q − h + p − h + q + p − = a − ( u, v, p, q ) = [ v − u + p + q ] (13) W uv h h + q − h + p h + q + p − = b + ( u, v, p, q, h ) (14)= [2 h ] [2( h + p + q )] [2( h + p )] [2( h + q )] [ u − v + q − p ] W uv h h + qh + p − h + q + p − = b − ( u, v, p, q, h ) (15)= [2 h ] [2( h + p + q )] [2( h + p )] [2( h + q )] [ u − v + p − q ] W uv h h + qh + p h + q + p − = c + ( u, v, p, q, h ) (16)= [2 p ] [2 q ] [2( h + p )] [2( h + q )] [ v − u + p + q + 2 h ] W uv h h + q − h + p − h + q + p − = c − ( u, v, p, q, h ) (17)= [2 p ] [2 q ] [2( h + p )] [2( h + q )] [ u − v + p + q + 2 h ]2.5. The Yang-Baxter equations.
For rapidities { u, v, w } , external fields { p, q, r } ,and non-negative integers { k, l, m, n, o } , the above weights satisfy the Yang-Baxterequations X j ≥ W uv h h + q − jh + p − o h + q + p − n W uw h + q − j h + q + r − lh + q + p − n h + q + r + p − m × W vw h h + r − kh + q − j h + r + q − l = X j ≥ W uv h + r − k h + r + q − lh + r + p − j h + r + q + p − m W uw h h + r − kh + p − o h + r + p − j × W vw h + p − o h + p + r − jh + p + q − n h + p + r + q − m (18) Proof.
This can be proved by direct computation using elliptic function identities,along the same lines as in [4]. For example, when { k, l, m, n, o } = { , , , , } , theYang-Baxter equation is O FODA, M WHEELER AND M ZUPARIC1 X j =0 W uv h h + q − jh + p − h + q + p − W uw h + q − j h + q + r − h + q + p − h + q + r + p − × W vw h h + rh + q − j h + r + q − = W uv h + r h + r + q − h + r + p − h + r + q + p − W uw h h + rh + p − h + r + p − × W vw h + p − h + p + r − h + p + q − h + p + r + q − (19)Using the expressions for the weights, we obtain c − ( u, v, p, q, h ) a + ( u, w, p, r ) b − ( v, w, q, r, h )+ b − ( u, v, p, q, h ) c − ( u, w, p, r, h + q ) c + ( v, w, q, r, h )= c − ( u, v, p, q, h + r ) b − ( u, w, p, r, h ) a + ( v, w, q, r ) (20)Writing the weights in terms of elliptic functions, one can eliminate commonfactors, and the proof of the equation reduces to proving[ u − v + p + q + 2 h ][ u − w + p + r ][ v − w + q − r ][2( h + q + r )]+ [ u − v + p − q ][ u − w + p + r + 2( h + q )][ w − v + q + r + 2 h ][2 r ]= [ u − v + p + q + 2( h + r )][ u − w + p − r ][ v − w + q + r ][2( h + q )] (21)which proceeds by noting that the ratio of the left-hand-side and right-hand-side isdoubly periodic and entire in u , and therefore a constant with respect to u . Setting u = v − p + q , the constant is found to be 1.2.6. Switching off the external fields.
Setting p = q = is equivalent toswitching off the external fields. This becomes clear by inspection of the vertexweights, which up to normalization become equal to those of Baxter’s solid-on-solid model at the free fermion point.2.7. The external fields tilt the heights.
One can think of the external fields p = and/or q = , as effectively tilting the heights of the lattice faces that theyflow through. This tilt is with respect to the line along which a field flows. Thiseffectively adds to or subtracts from the height differences that are the case in theabsence of external fields.2.8. The c + vertex. In discussions of DWBC’s and DWPF’s, the c + vertex, seeFigure , plays a special role: It is the DWPF for a 1 × LLIPTIC HEIGHT MODELS AND FACTORIZED DWPF 7 u , p v , qhh + p h + qh + q + p– Figure 3.
The Felderhof-type c + vertex. Domain wall boundary conditions (DWBC).
We define the DWBC’s asan expanded c + vertex, as in Figure : Given the external fields { p, q } and startingfrom h = h at the top-left corner, the boundary heights change by q j from left toright along the upper boundary, p i − − q j + 1 from right to left along the lower boundary, and − p i from bottom to topalong the left boundary. hh + p h + p , h + p ,L h + q ,L h + q ,L + p − h + q ,L + p , − h + q ,L + p ,L − Lh + q + p ,L − h + q , + p ,L − Figure 4.
Felderhof-type height domain wall boundary conditions. We use thenotation p i,j = P jk = i p k , etc. Domain wall partition function (DWPF).
The DWPF on an L × L lattice, Z L × L , is the sum over all weighted configurations that satisfy the DWBC.The weight of each configuration is the product of the weights of the vertices O FODA, M WHEELER AND M ZUPARIC Z L × L = X configurations Y vertices w ij (22)2.11. Line permuting vertices.
In proofs of DWPF’s two vertices play an im-portant role. These are the a -type vertices which can be used to permute adjacentflow lines. u , p v , qhh + p h + qh + q + p u , p v , qhh + p– h + q– h + q + p– Figure 5.
The Felderhof-type line permuting vertices a + and a − . Different zeros.
The weights of the line permuting vertices, [ u − v + p + q ]and [ v − u + p + q ], have different zeros. This is the property that will allow us toobtain the zeros of the DWPF and compute it in factorized form.2.13. Properties of the partition function.
The following four properties de-termine the partition function uniquely.2.13.1.
Property 1: Quasi-periodicity.
The partition function is entire in u andsatisfies the quasi-periodicity conditions Z L × L u + 2 , . . . , u L , { v } , { p } , { q } , h = ( − ) L Z L × L { u } , { v } , { p } , { q } , h (23) Z L × L u − i log( q ) π , . . . , u L , { v } , { p } , { q } , h =( − ) L q L exp − πi Lu + ( L − p − L X j =1 ( v j + q j ) − h × Z L × L { u } , { v } , { p } , { q } , h (24) Proof.
Since the weights are entire functions in the rapidities, it follows that Z L × L { u } , { v } , { p } , { q } , h is an entire function in u . To prove the quasi-periodicity conditions, we write the partition function in the form Z L × L { u } , { v } , { p } , { q } , h = L X n =1 P n u , { v } , p , { q } , h × (25) Q n u , . . . , u L , { v } , p , . . . , p L , { q } , h LLIPTIC HEIGHT MODELS AND FACTORIZED DWPF 9 where P n u , { v } , p , { q } , h = n − Y j =1 a + ( u , v j , p , q j ) c + u , v n , p , q n , h + n − X k =1 q k (26) × L Y j = n +1 b − ( u , v j , p , q j , h + j − X k =1 q k ) and Q n u , . . . , u L , { v } , p , . . . , p L , { q } , h does not depend on u . Using theexpressions for the weights, we have P n u + 2 , { v } , p , { q } , h = ( − ) L P n u , { v } , p , { q } , h (27) P n u − i log( q ) π , { v } , p , { q } , h = (28)( − ) L q L exp − πi Lu + ( L − p − L X j =1 ( v j + q j ) − h P n u , { v } , p , { q } , h from which the required property follows immediately.2.13.2. Property 2: Simple zeros.
The partition function has simple zeros at u = u j − p − p j , where j = 2 , . . . , L . Proof.
We multiply the partition function by a + ( u , u , p , p ), and use the Yang-Baxter equation to slide the inserted face through the lattice. u , p u , p u , p u , p Figure 6.
Inserting an a + vertex into the left boundary. It emerges as a − ( u , u , p , p ), and the order of the first two lattice rows isreversed. u , p u , p u , p u , p Figure 7.
Extracting an a − vertex from the right boundary. This is equivalent to the equation Z L × L { u } , { v } , { p } , { q } , h = a − ( u , u , p , p ) a + ( u , u , p , p ) × (29) Z L × L u , u , . . . , u L , { v } , p , p , . . . , p L , { q } , h Repeating this procedure on the second and third rows, and so on, we obtain Z L × L { u } , { v } , { p } , { q } , h = L Y j =2 a − ( u j , u , p j , p ) a + ( u j , u , p j , p ) × (30) Z L × L u , . . . , u L , u , { v } , p , . . . , p L , p , { q } , h which has the required simple zeros in the numerator.2.13.3. Property 3: A recursion relation.
The partition function satisfies the recur-sion relation Z L × L { u } , { v } , { p } , { q } , h (cid:12)(cid:12)(cid:12) u = v − p − q = c + ( v − p − q , v , p , q , h ) × L Y j =2 b + ( u j , v , p j , q , h + j − X k =1 p k ) b − ( v − p − q , v j , p , q j , h + j − X k =1 q k ) × Z ( L − × ( L − u , . . . , u L , v , . . . , v L , p , . . . , p L , q , . . . , q L , h + p + q − (31) Proof.
In any lattice configuration in the partition function sum, the top-left cornerof the lattice must be a + ( u , v , p , q ) or c + ( u , v , p , q , h ). Setting u = v − p − q in the partition function sets to zero all configurations with a + ( u , v , p , q ).The surviving configurations must have a top-left corner equal to c + , which fixesthe rest of the top row to b − , the rest of the first column to b + , and the remainder LLIPTIC HEIGHT MODELS AND FACTORIZED DWPF 11 of the lattice to Z ( L − × ( L − . The above recursion follows immediately from theseconsiderations.2.13.4. Property 4.
The partition function on a 1 × Z × ( u , v , p , q , h ) = c + ( u , v , p , q , h ) (32) Proof.
This follows from the definition of domain wall boundary conditions.2.14.
The partition function is uniquely determined.
Assume that Z ( n − × ( n − is uniquely determined by the above four properties, for some n ≥
2. From Property , Property and Theorem , we have Z n × n { u } , { v } , { p } , { q } , h = κ ( u , . . . , u n , { v } , { p } , { q } ) × [ n X j =1 ( v j − u j ) + n X j =1 ( p j + q j ) + 2 h ] n Y j =2 [ u − u j + p + p j ] (33)Property fully determines the coefficient κ in terms of Z ( n − × ( n − . Finally,since Z × is uniquely determined by Property , Z n × n is uniquely determined bythe four properties.2.15. The domain wall partition function.
The solution to the preceding fourproperties is given by Z L × L { u } , { v } , { p } , { q } , h = Q Lj =1 [2 p j ] [2 q j ] [2( h + P Lj =1 p j )] [2( h + P Lj =1 q j )] × [ L X j =1 ( v j − u j ) + L X j =1 ( p j + q j ) + 2 h ] Y ≤ j There are two possible choices of DWBC’s. Onecorresponds to an expanded c + , as in this work, and one to an ‘expanded’ c − vertex.The DWPF depends on the choice. The two expressions coincide for vanishingexternal fields, that is p i = q j = , and appropriate choices of the boundary heightvariables. 3. A Perk-Schultz-type gl (1 | height model In [3], Deguchi and Martin introduced elliptic height versions of the gl ( r +1 | s +1)trigonometric vertex models. In the following, we define DWBC’s and compute theDWPF in the gl (1 | 1) case. Since the analysis in this Section follows almost verbatimthat of Section , we will be brief and give just enough details where the two modelsdiffer. Notation, heights and restrictions. In this model, there are two squarelattices. A physical L × L lattice that the heights live on, and a target Z × Z latticethat the heights take values in. The target lattice is spanned by the unit vectors ˆ e µ , µ ∈ {− , +1 } , thus the height variables are 2-component vectors { h µ , h ν } . Heightson adjacent physical lattice corners are restricted to take values in adjacent pointson the target lattice. Height differences along ˆ e − and ˆ e +1 lead to different vertexweights. To each height vector h = { h µ , h ν } , we assign a scalar h µν = h µ + h ν + ω µν (35)where ω is an arbitrary constant antisymmetric complex 2 × e µ as well as ˆ e sign( µ ) to indicate the same unit vector.3.2. No external fields and the crossing parameter is a variable. Unlikethe previous Felderhof-type model, the Perk-Schultz-type model in this Section hasno external fields. The crossing parameter is left as a variable.3.3. The weights. The non-zero vertex weights are W uv h h + ˆ e µ h + ˆ e µ h + 2ˆ e µ = a µ ( u, v ) = [1 + µ ( u − v )][1] (36) W uv h h + ˆ e ν h + ˆ e µ h + ˆ e µ + ˆ e ν = b µ ( u, v ) = [ u − v ][ h µν − h µν ] , µ = ν (37) W uv h h + ˆ e µ h + ˆ e µ h + ˆ e µ + ˆ e ν = c µ ( u, v ) = [ h µν − ( u − v )][ h µν ] , µ = ν (38)The variables h µν on the right hand sides of Equations and are the scalarsassigned to the heights at the upper left corners of the corresponding vertices.3.4. The Yang-Baxter equations. The weights satisfy the following Yang-Baxterequations [3]. X g ∈ Z W u u b ga f W u u g df e W u u b cg d = X g ∈ Z W u u a gf e W u u b ca g W u u c dg e (39)3.5. The c + vertex. We take the c + vertex to be as in Figure . u vhh +ˆ e + h +ˆ e + h +ˆ e + + ˆ e − Figure 8. A general face configuration. LLIPTIC HEIGHT MODELS AND FACTORIZED DWPF 13 hh + ˆ e + h + 2ˆ e + h + L ˆ e + h + L ˆ e + h + L ˆ e + + ˆ e − h + L ˆ e + + 2ˆ e − h + L ˆ e − + L ˆ e + h + L ˆ e + + ˆ e − h + L ˆ e + + 2ˆ e − Figure 9. Perk-Schultz-type height domain wall boundary conditions. The domain wall boundary conditions. We choose the DWBC as in Fig-ure . In other words, starting from the lower left corner, all height changes alongthe left boundary are of type ˆ e +1 , along the upper boundary they are of type ˆ e − ,along the right boundary they are of type ˆ e +1 , then along the lower boundary theyare of type ˆ e − .3.7. The line-permuting vertices. We take the line permuting vertices to be asin Figure . Their weights have different zeros leading to a factorization of theDWPF, just as in Section . u vhh +ˆ e + h +ˆ e + h +2ˆ e + u vhh +ˆ e − h +ˆ e − h +2ˆ e − Figure 10. The Perk-Schultz-type line permuting vertices a + and a − . The DWPF. Having defined the model, the derivation of the correspondingDWPF proceeds precisely in analogy with that in Section . Based on the quasi-periodicity properties of the partition function, we propose a factorization in termsof the [ u ] functions. We obtain the zeros by permuting adjacent flow lines, usingthe line permuting vertices, obtain a recursion relation the DWPF satisfies and an initial condition. The DWPF is uniquely determined, and the following expressionsatisfies all the conditions. Z L × L =[ h + ( L − − P Lk =1 ( u k − v k )][ h + ( L − Y ≤ i 1) model.4. Remarks The point of this work is to give examples of domain wall partition functions inelliptic and/or height models. We restricted our attention to models that are non-invariant under state variable conjugation, which greatly simplified the problem.Because of the fermionic nature of the models discussed in this paper, there isno interesting applications of their DWPF’s to enumerations of alternating signmatrices or related objects that we are aware of.It is highly likely that all models that are non-invariant under some form of statevariable conjugation, that allows factorization, are fermionic (as the Felderhof-typemodel of Section ) or contain fermions that play an essential role in the definition ofthe DWBC’s (as the Perk-Schultz-type model of Section ). As such, these modelsare unrepresentative of the general case. However, they are non-trivial and wehope that one can learn something by extending our results to compute correlationfunctions. Acknowledgements OF would like to thank Professors R J Baxter, T Deguchi and M Jimbo fordiscussions, and P Bouwknegt, T Guttmann and T Miwa for hospitality at ANU,MASCOS and Kyoto University, while this work was in progress. MW and MZ aresupported by an Australian Postgraduate Award (APA). References [1] O Foda, M Wheeler and M Zuparic , J Stat Mech (2007) P10016 arXiv:0709.4540 [2] V E Korepin and O I Patu , XXX spin chain: From Bethe solution to open problems , arXiv:cond-mat/0701491 [3] T Deguchi and P P Martin Int J Mod Phys A7 , Suppl. (1992) 165-196[4] R J Baxter Exactly solved models ,[5] T Deguchi and A Akutsu , J Phys Soc of Japan (1993) 19–35.[6] T Deguchi and A Akutsu , J Phys Soc of Japan (1991) 4051–4059. Department of Mathematics and Statistics, University of Melbourne, Parkville, Vic-toria 3010, Australia. E-mail address ::