On Spin Models,Triply Regular Association Schemes,and Duality

Motivated by the construction of invariants of links in 3-space, we study spin models on graphs for which all edge weights (considered as matrices) belong to the Bose-Mesner algebra of some association scheme. We show that for series-parallel graphs the computation of the partition function can be performed by using series-parallel reductions of the graph appropriately coupled with operations in the Bose-Mesner algebra. Then we extend this approach to all plane graphs by introducing star-triangle transformations and restricting our attention to a special class of Bose-Mesner algebras which we call exactly triply regular. We also introduce the following two properties for Bose-Mesner algebras. The planar duality property (defined in the self-dual case) expresses the partition function for any plane graph in terms of the partition function for its dual graph, and the planar reversibility property asserts that the partition function for any plane graph is equal to the partition function for the oppositely oriented graph. Both properties hold for any Bose-Mesner algebra if one considers only series-parallel graphs instead of arbitrary plane graphs. We relate these notions to spin models for link invariants, and among other results we show that the Abelian group Bose-Mesner algebras have the planar duality property and that for self-dual Bose-Mesner algebras, planar duality implies planar reversibility. We also prove that for exactly triply regular Bose-Mesner algebras, to check one of the above properties it is sufficient to check it on the complete graph on four vertices. A number of applications, examples and open problems are discussed.     

Spin Models of Index 2 and Hadamard Models

A spin model (for link invariants) is a square matrix W with non-zero complex entries which satisfies certain axioms. Recently it was shown that ^t WW ^–1 is a permutation matrix (the order of this permutation matrix is called the index of W), and a general form was given for spin models of index 2. Moreover, new spin models, called non-symmetric Hadamard models, were constructed. In the present paper, we classify certain spin models of index 2, including non-symmetric Hadamard models.     

General Form of Non-Symmetric Spin Models

A spin model (for link invariants) is a square matrix W with non-zero complex entries which satisfies certain axioms. Recently (Jaeger and Nomura, J. Alg. Combin. 10 (1999), 241–278) it was shown that ^t WW ^–1 is a permutation matrix (the order of this permutation matrix is called the index of W), and a general form was given for spin models of index 2. In the present paper, we generalize this general form to an arbitrary index m. In particular, we give a simple form of W when m is a prime number.     

Spin Models and Strongly Hyper-Self-Dual Bose-Mesner Algebras

We introduce the notion of hyper-self-duality for Bose-Mesner algebras as a strengthening of formal self-duality. Let denote a Bose-Mesner algebra on a finite nonempty set X. Fix p X, and let and denote respectively the dual Bose-Mesner algebra and the Terwilliger algebra of with respect to p. By a hyper-duality of , we mean an automorphism of such that for all ; and is a duality of . is said to be hyper-self-dual whenever there exists a hyper-duality of . We say that is strongly hyper-self-dual whenever there exists a hyper-duality of which can be expressed as conjugation by an invertible element of . We show that Bose-Mesner algebras which support a spin model are strongly hyper-self-dual, and we characterize strong hyper-self-duality via the module structure of the associated Terwilliger algebra.     

Symmetric Versus Non-Symmetric Spin Models for Link Invariants

We study spin models as introduced in [20]. Such a spin model can be defined as a square matrix satisfying certain equations, and can be used to compute an associated link invariant. The link invariant associated with a symmetric spin model depends only trivially on link orientation. This property also holds for quasi-symmetric spin models, which are obtained from symmetric spin models by certain gauge transformations preserving the associated link invariant. Using a recent result of [16] which asserts that every spin model belongs to some Bose-Mesner algebra with duality, we show that the transposition of a spin model can be realized by a permutation of rows. We call the order of this permutation the index of the spin model. We show that spin models of odd index are quasi-symmetric. Next, we give a general form for spin models of index 2 which implies that they are associated with a certain class of symmetric spin models. The symmetric Hadamard spin models of [21] belong to this class and this leads to the introduction of non-symmetric Hadamard spin models. These spin models give the first known example where the associated link invariant depends non-trivially on link orientation. We show that a non-symmetric Hadamard spin model belongs to a certain triply regular Bose-Mesner algebra of dimension 5 with duality, and we use this to give an explicit formula for the associated link invariant involving the Jones polynomial.     

Characterization of SDP Designs That Yield Certain Spin Models

We characterize the SDP designs that give rise to four-weight spin models with two values. We prove that the only such designs are the symplectic SDP designs. The proof involves analysis of the cardinalities of intersections of four blocks.AMS classification:05B20, 05E30     

Spin Models on Finite Cyclic Groups

The concept of spin model is due to V. F. R. Jones. The concept of nonsymmetric spin model, which generalizes that of the original (symmetric) spin model, is defined naturally. In this paper, we first determine the diagonal matrices T satisfying the modular invariance or the quasi modular invariance property, i.e., or (respectively), for the character table P of the group association scheme of a cyclic group G of order m. Then we show that a (symmetric or nonsymmetric) spin model on G is constructed from each of the matrices T satisfying the modular or quasi modular invariance property.     

On Four-Weight Spin Models and their Gauge Transformations

We study the four-weight spin models (W₁, W₂, W₃, W₄) introduced by Eiichi and Etsuko Bannai (Pacific J. of Math, to appear). We start with the observation, based on the concept of special link diagram, that two such spin models yield the same link invariant whenever they have the same pair (W₁, W₃), or the same pair (W₂, W₄). As a consequence, we show that the link invariant associated with a four-weight spin model is not sensitive to the full reversal of orientation of a link. We also show in a similar way that such a link invariant is invariant under mutation of links.Next, we give an algebraic characterization of the transformations of four-weight spin models which preserve W₁, W₃ or preserve W₂, W₄. Such gauge transformations correspond to multiplication of W₂, W₄ by permutation matrices representing certain symmetries of the spin model, and to conjugation of W₁, W₃ by diagonal matrices. We show for instance that up to gauge transformations, we can assume that W₁, W₃ are symmetric.Finally we apply these results to two-weight spin models obtained as solutions of the modular invariance equation for a given Bose-Mesner algebra B and a given duality of B. We show that the set of such spin models is invariant under certain gauge transformations associated with the permutation matrices in B. In the case where B is the Bose-Mesner algebra of some Abelian group association scheme, we also show that any two such spin models (which generalize those introduced by Eiichi and Etsuko Bannai in J. Alg. Combin. 3 (1994), 243–259) are related by a gauge transformation. As a consequence, the link invariant associated with such a spin model depends only trivially on the link orientation.     

Symmetric Designs Attached to Four-Weight Spin Models

H. Guo and T. Huang studied the four-weight spin models (X, W ₁, W ₂, W ₃, W ₄;D) with the property that the entries of the matrix W ₂ (or equivalently W ₄) consist of exactly two distinct values. They found that such spin models are always related to symmetric designs whose derived design with respect to any block is a quasi symmetric design. In this paper we show that such a symmetric design admits a four-weight spin model with exactly two values on W ₂ if and only if it has some kind of duality between the set of points and the set of blocks. We also give some examples of parameters of symmetric designs which possibly admit four-weight spin models with exactly two values on W ₂.     

Bose-Mesner Algebras Related to Type II Matrices and Spin Models

A type II matrix is a square matrixW with non-zero complex entries such that the entrywise quotient of any two distinct rows of W sums to zero. Hadamard matrices and character tables of abelian groups are easy examples, and other examples called spin models and satisfying an additional condition can be used as basic data to construct invariants of links in 3-space. Our main result is the construction, for every type II matrix W, of a Bose-Mesner algebra N(W) , which is a commutative algebra of matrices containing the identity I, the all-one matrix J, closed under transposition and under Hadamard (i.e., entrywise) product. Moreover, ifW is a spin model, it belongs to N(W). The transposition of matrices W corresponds to a classical notion of duality for the corresponding Bose-Mesner algebrasN(W) . Every Bose-Mesner algebra encodes a highly regular combinatorial structure called an association scheme, and we give an explicit construction of this structure. This allows us to compute N(W) for a number of examples.     

On Spin Models, Modular Invariance, and Duality

A spin model is a triple (X, W ⁺, W ^–), where W ⁺ and W ^– are complex matrices with rows and columns indexed by X which satisfy certain equations (these equations allow the construction of a link invariant from(X, W ⁺, W ^–) ). We show that these equations imply the existence of a certain isomorphism between two algebras and associated with (X, W ⁺, W ^–) . When is the Bose-Mesner algebra of some association scheme, and is a duality of . These results had already been obtained in [15] when W ⁺, W ^– are symmetric, and in [5] in the general case, but the present proof is simpler and directly leads to a clear reformulation of the modular invariance property for self-dual association schemes. This reformulation establishes a correspondence between the modular invariance property and the existence of spin models at the algebraic level. Moreover, for Abelian group schemes, spin models at the algebraic level and actual spin models coincide. We solve explicitly the modular invariance equations in this case, obtaining generalizations of the spin models of Bannai and Bannai [3]. We show that these spin models can be identified with those constructed by Kac and Wakimoto [20] using even rational lattices. Finally we give some examples of spin models at the algebraic level which are not actual spin models.     

Quasifree States in Some One-Dimensional Quantum Spin Models

We use numerical methods to investigate the SU _q(N) Perk–Schultz spin chain at the special quantum parameter value q=–e ^i/N . We discover simple laws applicable to a considerable part of the Hamiltonian spectrum, which in particular contains the energy of the ground state and the nearest excitations. The phenomenological formulas obtained resemble formulas for the spectrum of the free-fermion model. We formulate several hypotheses, some of which can be justified by constructing exact solutions of the system of Bethe-ansatz equations for finite-length chains. We obtain two sets of solutions of these equations. The first corresponds to the special value of the quantum parameter q and, in particular, describes the model ground state, which is antiferromagnetic. The second set of solutions describes a part of the spectrum belonging to the sectors where the numbers n _i of particles of different types (i=0,1,...,N–1) do not exceed unity for all the types except one. For this set, we obtain a simple spectrum at arbitrary values of q. It is hypothesized that this spectrum and the solutions of the Bethe-ansatz equations found in a closed form are intimately related to the existence of a special eigenstate for the transfer matrix of the auxiliary inhomogeneous SU _q(N–1) vertex model that is involved in constructing the system of Bethe-ansatz equations of a matrioshka structure. Indirect arguments based on combinatorial properties of the wave function of the relevant state are given to support this hypothesis.     

Duality Maps of Finite Abelian Groups and Their Applications to Spin Models

Duality maps of finite abelian groups are classified. As a corollary, spin models on finite abelian groups which arise from the solutions of the modular invariance equations are determined as tensor products of indecomposable spin models. We also classify finite abelian groups whose Bose-Mesner algebra can be generated by a spin model.     

Twisted Frobenius Extensions of Graded Superrings

We define twisted Frobenius extensions of graded superrings. We develop equivalent definitions in terms of bimodule isomorphisms, trace maps, bilinear forms, and dual sets of generators. The motivation for our study comes from categorification, where one is often interested in the adjointness properties of induction and restriction functors. We show that A is a twisted Frobenius extension of B if and only if induction of B-modules to A-modules is twisted shifted right adjoint to restriction of A-modules to B-modules. A large (non-exhaustive) class of examples is given by the fact that any time A is a Frobenius graded superalgebra, B is a graded subalgebra that is also a Frobenius graded superalgebra, and A is projective as a left B-module, then A is a twisted Frobenius extension of B.     

An Algebra Associated with a Spin Model

To each symmetric n × n matrix W with non-zero complex entries, we associate a vector space N, consisting of certain symmetric n × n matrices. If W satisfies then N becomes a commutative algebra under both ordinary matrix product and Hadamard product (entry-wise product), so that N is the Bose-Mesner algebra of some association scheme. If W satisfies the star-triangle equation: then W belongs to N. This gives an algebraic proof of Jaeger's result which asserts that every spin model which defines a link invariant comes from some association scheme.     

Homogeneity of a Distance-Regular Graph Which Supports a Spin Model

A spin model is a square matrix that encodes the basic data for a statistical mechanical construction of link invariants due to V.F.R. Jones. Every spin model W is contained in a canonical Bose-Mesner algebra (W). In this paper we study the distance-regular graphs whose Bose-Mesner algebra satisfies W (W). Suppose W has at least three distinct entries. We show that is 1-homogeneous and that the first and the last subconstituents of are strongly regular and distance-regular, respectively.     

Exact solutions and mixing in an algebraic dynamical system

Let be an n×n matrix with entries a_ij in the field . We consider two involutive operations on these matrices: the matrix inverse I: ^–1 and the entry-wise or Hadamard inverse J: a_ij a _ij ^–1 . We study the algebraic dynamical system generated by iterations of the product J. I. We construct the complete solution of this system for n 4. For n = 4, it is obtained using an ansatz in theta functions. For n 5, the same ansatz gives partial solutions. They are described by integer linear transformations of the product of two identical complex tori. As a result, we obtain a dynamical system with mixing described by explicit formulas.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 143, No. 1, pp. 131–149, April, 2005.     

The Terwilliger Algebra of a Distance-Regular Graph that Supports a Spin Model

Let denote a distance-regular graph with vertex set X, diameter D 3, valency k 3, and assume supports a spin model W. Write W = _{i = 0}^D t_i A_i where A_i is the ith distance-matrix of . To avoid degenerate situations we assume is not a Hamming graph and t_i {t₀, –t₀ } for 1 i D. In an earlier paper Curtin and Nomura determined the intersection numbers of in terms of D and two complex parameters and q. We extend their results as follows. Fix any vertex x X and let T = T(x) denote the corresponding Terwilliger algebra. Let U denote an irreducible T-module with endpoint r and diameter d. We obtain the intersection numbers c_i(U), b_i(U), a_i(U) as rational expressions involving r, d, D, and q. We show that the isomorphism class of U as a T-module is determined by r and d. We present a recurrence that gives the multiplicities with which the irreducible T-modules appear in the standard module. We compute these multiplicites explicitly for the irreducible T-modules with endpoint at most 3. We prove that the parameter q is real and we show that if is not bipartite, then q > 0 and is real.AMS 2000 Subject Classification: Primary 05E30     

Backlund Loop Algebras for Compact and Noncompact Nonlinear Spin Models in 2+1 Dimensions

We solve the Backlund problem for both the compact and noncompact versions of the Ishimori (2+1)-dimensional nonlinear spin model. In particular, we realize the arising Backlund algebra in the form of an infinite-dimensional loop Lie algebra of the Kac-Moody type.This note is part of joint research work with E. Winterroth.__________Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 144, No. 1, pp. 153–161, July, 2005.