algebraic reduction of the ising model we consider the ising model on a cylindrical lattice of l columns with fixed spin boundary conditions on the top and bottom rows the spontaneous magnetization can be written in terms of partition functions on this lattice we show how we can use the clifford algebra of kaufman to write these partition functions in terms of l by l determinants and then further reduce them to m by m determinants where m is approximately l 2 in this form the results can be compared with those of the ising case of the superintegrable chiral potts model they point to a way of calculating the spontaneous magnetization of that more general model algebraically

a twist on chiral potts we show that the chiral potts model may be formulated so that the rapidity lines carry a second integer variable an increment or twist in each bond crossing it this modification does not affect those properties of the chiral potts model which lead to integrability since it is equivalent to one of the automorphisms allowed for in the theory in particular transfer matrices still form commuting families and still satisfy hierarchies of functional equations surprisingly the superintegrable case with twists retains the special algebraic properties which lead to its ising like spectra the formalism should be useful for considering systems with twisted boundary conditions or embedded interfaces

on the shape of a tridiagonal pair abstract let k denote a field and let v denote a vector space over k with finite positive dimension we consider a pair of linear transformations a v v and a v v that satisfy the following conditions i each of a a is diagonalizable ii there exists an ordering v i i 0 d of the eigenspaces of a such that a v i v i 1 v i v i 1 for 0 i d where v 1 0 and v d 1 0 iii there exists an ordering v i i 0 of the eigenspaces of a such that av i v i 1 v i v i 1 for 0 i where v 1 0 and v 1 0 iv there is no subspace w of v such that aw w a w w w 0 w v we call such a pair a tridiagonal pair on v it is known that d and for 0 i d the dimensions of v i v i v d i v d i coincide we denote this common dimension by i in this paper we prove that i 0 d i for 0 i d it is already known that 0 1 if k is algebraically closed

structure of certain chebyshev type polynomials in onsager s algebra representation in this report we present a systematic account of mathematical structures of certain special polynomials arisen from the energy study of the superintegrable n state chiral potts model with a finite number of sizes the polynomials of low lying sectors are represented in two different forms one of which is directly related to the energy description of superintegrable chiral potts z n spin chain via the representation theory of onsager s algebra both two types of polynomials satisfy some n 1 term recurrence relations and nth order differential equations polynomials of one kind reveal certain chebyshev like properties here we provide a rigorous mathematical argument for cases n 2 3 and further raise some mathematical conjectures on those special polynomials for a general n

leonard pairs in classical mechanics leonard pairs lp are matrices with the property of mutual tri diagonality we introduce and study a classical analogue of lp we show that corresponding classical leonard dynamical variables satisfy non linear relations of the aw type with respect to poisson brackets

chiral potts model as a descendant of the six vertex model we observe that then state integrable chiral potts model can be considered as a part of some new algebraic structure related to the six vertex model as a result we obtain a functional equation which is supposed to determine all the eigenvalues of the chiral potts model transfer matrix

reduced bloch mode expansion for periodic media band structure calculations reduced bloch mode expansion rbme is presented for fast periodic media band structure calculations the expansion employs a natural basis composed of a selected reduced set of bloch eigenfunctions the reduced basis is selected within the irreducible brillouin zone at high symmetry points determined by the medium s crystal structure and group theory and possibly at additional related points at each of the reciprocal lattice selection points a number of bloch eigenfunctions are selected up to the frequency energy range of interest for the band structure calculations as it is common to initially discretize the periodic unit cell and solution field using some choice of basis rbme is practically a secondary expansion that uses a selected set of bloch eigenvectors such expansion therefore keeps and builds on any favourable attributes a primary expansion approach might exhibit being in line with the well known concept of modal analysis the proposed approach maintains accuracy while reducing the computation time by up to two orders of magnitudes or more depending on the size and extent of the calculations results are presented for phononic photonic and electronic band structures

fusion operators in the generalized τ 2 model and root of unity symmetry of the xxz spin chain of higher spin we construct the fusion operators in the generalized 2 model using the fused l operators and verify the fusion relations with the truncation identity the algebraic bethe ansatz discussion is conducted on two special classes of 2 which include the superintegrable chiral potts model we then perform the parallel discussion on the xxz spin chain at roots of unity and demonstrate that the sl2 loop algebra symmetry exists for the root of unity xxz spin chain with a higher spin where the evaluation parameters for the symmetry algebra are identified by the explicit fabricius mccoy current for the bethe states parallels are also drawn to the comparison with the superintegrable chiral potts model

a classification of sharp tridiagonal pairs let f denote a field and let v denote a vector space over f with finite positive dimension we consider a pair of linear transformations a v v and a v v that satisfy the following conditions i each of a a is diagonalizable ii there exists an ordering vi i 0d of the eigenspaces of a such that a vi vi 1 vi vi 1 for 0 i d where v 1 0 and vd 1 0 iii there exists an ordering vi i 0 of the eigenspaces of a such that avi vi 1 vi vi 1 for 0 i where v 1 0 and v 1 0 iv there is no subspace w of v such that aw w a w w w 0 w v we call such a pair a tridiagonal pair on v it is known that d and for 0 i d the dimensions of vi vd i vi vd i coincide the pair a a is called sharp whenever dimv0 1 it is known that if f is algebraically closed then a a is sharp in this paper we classify up to isomorphism the sharp tridiagonal pairs as a corollary we classify up to isomorphism the tridiagonal pairs over an algebraically closed field we obtain these classifications by proving the conjecture

the structure of quotients of the onsager algebra by closed ideals we study the onsager algebra from the ideal theoretic point of view a complete classification of closed ideals and the structure of quotient algebras are obtained we also discuss the solvable algebra aspect of the onsager algebra through the use of formal lie algebras

two linear transformations each tridiagonal with respect to an eigenbasis of the other let k denote a field and let v denote a vector space over k with finite positive dimension we consider a pair of linear transformations a v v and a v v satisfying both conditions below 1 i there exists a basis for v with respect to which the matrix representing a is diagonal and the matrix representing a is irreducible tridiagonal 2 ii there exists a basis for v with respect to which the matrix representing a is diagonal and the matrix representing a is irreducible tridiagonal we call such a pair a leonard pair on v refining this notion a bit we introduce the concept of a leonard system we give a complete classification of leonard systems integral to our proof is the following result we show that for any leonard pair a a on v there exists a sequence of scalars taken from k such that both 0 a a2a aa a a a2 aa a a a 0 a a 2a a aa aa 2 a a aa a where r s means rs sr the sequence is uniquely determined by the leonard pair if the dimension of v is at least 4 we conclude by showing how leonard systems correspond to q racah and related polynomials from the askey scheme

superintegrable chiral potts model thermodynamic properties an inverse model and a simple associated hamiltonian the partition function of then state superintegrable chiral potts model is obtained exactly and explicitly if not completely rigorously for a finite lattice with particular boundary conditions this yields the bulk and surface free energies and horizontal and vertical correlation lengths and interfacial tensions the critical exponents are 1 2 n hor hor 2 n and vert vert 1 and the finite size corrections are obtained at criticality the eigenvalue spectrum of the column to column transfer matrix is that of a direct product ofn byn matrices inverting this matrix gives a related solvable model which is a generalization of the free fermion model the associated hamiltobian has a very simple form suggesting there may be a more direct algebraic method perhaps a generalized clifford algebra for obtaining its eigenvalues

on the depth of the associated graded ring of a filtration abstract let f be a filtration of a cohen macaulay ring such that we can define its associated graded ring g f we will give a suitable condition on f which enables us to estimate the depth of g f it generalizes several known results on associated graded rings of ideals

the early history of the integrable chiral potts model and the odd even problem in the first part of this paper i shall discuss the round about way of how the integrable chiral potts model was discovered about 30 years ago as there should be more higher genus models to be discovered this might be of interest in the second part i shall discuss some quantum group aspects especially issues of odd versus even n related to the serre relations conjecture in our quantum loop subalgebra paper of 5 years ago and how we can make good use of coproducts also borrowing ideas of drinfeld jimbo deguchi fabricius mccoy and nishino

the tetrahedron algebra the onsager algebra and the sl2 loop algebra let k denote a field with characteristic 0 and let t denote an indeterminate we give a presentation for the three point loop algebra sl2 k t t 1 t 1 1 via generators and relations this presentation displays s4 symmetry using this presentation we obtain a decomposition of the above loop algebra into a direct sum of three subalgebras each of which is isomorphic to the onsager algebra

some remarks on a generalization of the superintegrable chiral potts model the spontaneous magnetization of a two dimensional lattice model can be expressed in terms of the partition function w of a system with fixed boundary spins and an extra weight dependent on the value of a particular central spin for the superintegrable case of the chiral potts model with cylindrical boundary conditions w can be expressed in terms of reduced hamiltonians h and a central spin operator s we conjectured in a previous paper that w can be written as a determinant similar to that of the ising model here we generalize this conjecture to any hamiltonians that satisfy a more general onsager algebra and give a conjecture for the elements of s

onsager s algebra and the dolan grady condition in the non self dual case in this paper it is shown that a pair of operators that satisfy the dolan grady conditions generate an onsager algebra even in the case that they are not self dual briefly considered will be the implications for the spectra of superintegrable spin chains in various sectors

higher rank classical analogs of the askey wilson algebra from the sln onsager algebra the sl n onsager algebra has been introduced by uglov and ivanov in 1995 in this letter a frt presentation of the sl n onsager algebra is given its current algebra and commutative subalgebra are constructed certain quotients of the sl n onsager algebra are then considered which produce classical analogs of higher rank extensions of the askey wilson algebra as examples the cases n 3 and n 4 are described in details

identities in the superintegrable chiral potts model we present proofs for a number of identities that are needed to study the superintegrable chiral potts model in the q 0 sector

eigenvectors in the superintegrable model i sl2 generators in order to calculate correlation functions of the chiral potts model one only needs to study the eigenvectors of the superintegrable model here we start this study by looking for eigenvectors of the transfer matrix of the periodic 2 tq model which commutes with the chiral potts transfer matrix we show that the degeneracy of the eigenspace of 2 tq in the q 0 sector is 2r with r n 1 l n when the size of the transfer matrix l is a multiple of n we introduce chiral potts model operators different from the more commonly used generators of quantum group from these we can form the generators of a loop algebra for this algebra we then use the roots of the drinfeld polynomial to give new explicit expressions for the generators representing the loop algebra as the direct sum of r copies of the simple algebra

multi particle structure in the zn chiral potts models the authors calculate the lowest translationally invariant levels of the z3 and z4 symmetrical chiral potts quantum chains using numerical diagonalization of the hamiltonian for n or 12 and n or 10 sites respectively and extrapolating n to infinity in the high temperature massive phase they find that the pattern of the low lying zero momentum levels can be explained assuming the existence of n 1 particles carrying zn charges q 1 n 1 mass mq and their scattering states in the superintegrable case the masses of the n 1 particles become proportional to their respective charges mq qm1 exponential convergence in n is observed for the single particle gaps while power convergence is seen for the scattering levels they also verify that qualitatively the same pattern appears for the self dual and integrable cases for general zn they show that the energy momentum relations of the particles show a parity nonconservation asymmetry which for very high temperatures is exclusive due to the presence of a macroscopic momentum pm 1 2q n phi where phi is the chiral angle and q is the zn charge of the respective particle

transfer matrix eigenvectors of the baxter bazhanov stroganov τ2 model for n 2 we find a representation of the row to row transfer matrix of the baxter bazhanov stroganov 2 model for n 2 in terms of an integral over two commuting sets of grassmann variables using this representation we explicitly calculate transfer matrix eigenvectors and normalize them it is also shown how form factors of the model can be expressed in terms of determinants and inverses of certain toeplitz matrices

the baxter bazhanov stroganov model separation of variables and the baxter equation the baxter bazhanov stroganov model also known as the 2 model has attracted much interest because it provides a tool for solving the integrable chiral potts model it can be formulated as a face spin model or via cyclic l operators using the latter formulation and the sklyanin kharchev lebedev approach we give the explicit derivation of the eigenvectors of the component bn of the monodromy matrix for the fully inhomogeneous chain of finite length for the periodic chain we obtain the baxter t q equations via separation of variables the functional relations for the transfer matrices of the 2 model guarantee nontrivial solutions to the baxter equations for the n 2 case which is the free fermion point of a generalized ising model the baxter equations are solved explicitly

spin operator matrix elements in the superintegrable chiral potts quantum chain we derive spin operator matrix elements between general eigenstates of the superintegrable n symmetric chiral potts quantum chain of finite length our starting point is the extended onsager algebra recently proposed by baxter for each pair of spaces onsager sectors of the irreducible representations of the onsager algebra we calculate the spin matrix elements between the eigenstates of the hamiltonian of the quantum chain in factorized form up to an overall scalar factor this factor is known for the ground state onsager sectors for the matrix elements between the ground states of these sectors we perform the thermodynamic limit and obtain the formula for the order parameters for the ising quantum chain in a transverse field n 2 case the factorized form for the matrix elements coincides with the corresponding expressions obtained recently by the separation of variables method

the chiral potts models revisited we review some exact results obtained so far in the chiral potts models and translate these results into language more transparent to physicists so that experts in monte carlo calculations high and low temperature expansions and various other methods can use them we pay special attention to the interfacial tension r between thek state and thek r state by examining the ground states it is seen that the integrable line ends at a superwetting point on which the relation r r 1 is satisfied so that it is energetically neutral to have one interface or more we present also some partial results on the meaning of the integrable line for low temperatures where it lives in the nonwet regime we make baxter s exact results more explicit for the symmetric case by performing a bethe ansatz calculation with open boundary conditions we confirm a dilogarithm identity for the low temperature expansion which may be new we propose a new model for numerical studies this model has only two variables and exhibits commensurate and incommensurate phase transitions and wetting transitions near zero temperature it appears to be not integrable except at one point and at each temperature there is a point where it is almost identical with the integrable chiral potts model

onsager s algebra and superintegrability the author considers the irreducible representations of the onsager algebra and shows that for a finite system possessing such an algebra leads to an ising like structure in the spectra of associated hamiltonians and transfer matrices the chiral potts model is considered as an example for transfer in the diagonal direction it is known to be superintegrable for transfer in the principal direction a new superintegrable solution manifold is found

interfacial tension of the chiral potts model we calculate the interfacial tension of then state chiral potts model by solving the functional relations for the transfer matrices of the model with skewed boundary conditions our result is valid for the general physical model with positive boltzmann weights and at all subcritical temperatures the interfacial tension has been calculated previously for the superintegrable chiral potts model with skewed boundary conditions usingz invariance baxter has argued that the interfacial tension of this model should be the same as the interfacial tension of the general physical model we show that this is indeed the case

a conjecture for the superintegrable chiral potts model we adapt our previous results for the partition function of the superintegrable chiral potts model with open boundaries to obtain the corresponding matrix elements of e h where h is the associated hamiltonian the spontaneous magnetization r can be expressed in terms of particular matrix elements of e hs1re h where s1 is a diagonal matrix we present a conjecture for these matrix elements as an m by m determinant where m is proportional to the width of the lattice the author has previously derived the spontaneous magnetization of the chiral potts model by analytic means but hopes that this work will facilitate a more algebraic derivation similar to that of yang for the ising model

ising correlations and elliptic determinants correlation functions of the two dimensional ising model on the periodic lattice can be expressed in terms of form factors matrix elements of the spin operator in the basis of common eigenstates of the transfer matrix and translation operator free fermion structure of the model implies that any multiparticle form factor is given by the pfaffian of a matrix constructed from the two particle ones crossed two particle form factors can be obtained by inverting a block of the matrix of linear transformation induced on fermions by the spin conjugation we show that the corresponding matrix is of elliptic cauchy type and use this observation to solve the inversion problem explicitly non crossed two particle form factors are then obtained using theta functional interpolation formulas this gives a new simple proof of the factorized formulas for periodic ising form factors conjectured by a bugrij and one of the authors

spontaneous magnetization of the superintegrable chiral potts model calculation of the determinant d pq for the ising model the calculation of the spontaneous magnetization leads to the problem of evaluating a determinant yang did this by calculating the eigenvalues in the large lattice limit montroll potts and ward expressed it as a toeplitz determinant and used szeg s theorem this is almost certainly the route originally travelled by onsager for the corresponding problem in the superintegrable chiral potts model neither approach appears to work here we show that the determinant dpq can be expressed as that of a product of two cauchy like matrices one can then use the elementary exact formula for the cauchy determinant one of course regains the known result originally conjectured in 1989

on the depth of the associated graded ring of an ideal abstract let r be a local cohen macaulay ring let i be an r ideal and let g be the associated graded ring of i we give an estimate for the depth of g when g fails to be cohen macaulay we assume that i has a small reduction number and sufficiently good residual intersection properties and satisfies local conditions on the depth of some powers the main theorem unifies and generalizes several known results we also give conditions that imply the serre properties of the blow up rings

csos models descending from chiral potts models degeneracy of the eigenspace and loop algebra monodromy matrices of the model are known to satisfy a yang baxter equation with a six vertex r matrix as the intertwiner the commutation relations of the elements of the monodromy matrices are completely determined by this r matrix we show the reason why in the superintegrable case the eigenspace is degenerate but not in the general case we then show that the eigenspaces of special csos models descending from the chiral potts model are also degenerate the existence of an quantum loop algebra or subalgebra in these models is established by showing that the serre relations hold for the generators the highest weight polynomial or the drinfeld polynomial of the representation is obtained by using the method of baxter for the superintegrable case as a byproduct the eigenvalues of all such csos models are given explicitly

the order parameter of the chiral potts model an outstanding problem in statistical mechanics is the order parameter of the chiral potts model an elegant conjecture for this was made in 1983 it has since been successfully tested against series expansions but there is as yet no proof of the conjecture here we show that if one makes a certain analyticity assumption similar to that used to derive the free energy then one can indeed verify the conjecture the method is based on the broken rapidity line approach pioneered by jimbo et al j phys a 26 2199 2210 1993

sl n onsager s algebra and integrability we define ansl n analog of onsager s algebra through a finite set of relations that generalize the dolan grady defining relations for the original onsager s algebra this infinite dimensional lie algebra is shown to be isomorphic to a fixed point subalgebra ofsl n loop algebra with respect to a certain involution as the consequence of the generalized dolan grady relations a hamiltonian linear in the generators ofsl n onsager s algebra is shown to posses an infinite number of mutually commuting integrals of motion

chiral potts model with skewed boundary conditions we obtain the transfer matrix functional relations for the chiral potts model with skewed boundary conditions and find that they are the same as for periodic boundary conditions but with modified selection rules as a start toward calculating the interfacial tension in general we here evaluate it in a low temperature limit performing a bethe ansatz type calculation finally we specialize the relations to the superintegrable case verifying the ansatz proposed by albertiniet al

mock tridiagonal systems we introduce the notion of a mock tridiagonal system this is a generalization of a tridiagonal system in which the irreducibility assumption is replaced by a certain nonvanishing condition we show how mock tridiagonal systems can be used to construct tridiagonal systems that meet certain specifications this paper is part of our ongoing project to classify the tridiagonal systems up to isomorphism

form factors in the baxter bazhanov stroganov model i norms and matrix elements we continue our investigation of the baxter bazhanov stroganov model using the method of separation of variables 1 in this paper we calculate the norms and matrix elements of a local spin operator between eigenvectors of the auxiliary problem for the norm the multiple sums over the intermediate states are performed explicitly in the case n 2 we solve the baxter equation and obtain form factors of the spin operator of the periodic ising model on a finite lattice

the tetrahedron algebra and its finite dimensional irreducible modules abstract recently terwilliger and the present author found a presentation for the three point sl 2 loop algebra via generators and relations to obtain this presentation we defined a lie algebra by generators and relations and displayed an isomorphism from to the three point sl 2 loop algebra in this paper we classify the finite dimensional irreducible modules

integrability test for spin chains we examine a simple heuristic test of integrability for quantum chains this test is applied to a variety of systems including isotropic spin 1 models with nearest neighbour interaction potts type inhomogeneous spin 1 2 chains and a multiparameter family of spin 1 2 models generalizing the xyz chain with next to nearest neighbour interactions and bond alternation within the latter family we determine all the integrable models with an o 2 symmetry

cohen macaulay graded rings associated to ideals let g i be the associated graded ring of an ideal i in a cohen macaulay local ring a we give a sufficient condition for g i to be a cohen macaulay ring it is described in terms of the depths of a i n for finitely many n the reduction numbers of i q for certain prime ideals q and the artin nagata property of i in the sense of ulrich contemp math vol 159 pp 373 400 the main theorem unifies several results which are already known in this aspect of the theory although the statement of the main theorem is rather complicated we restate the conditions under special situations so that they are practical and simple

a bispectral q hypergeometric basis for a class of quantum integrable models for the class of quantum integrable models generated from the q onsager algebra a basis of bispectral multivariable q orthogonal polynomials is exhibited in the first part it is shown that the multivariable askey wilson polynomials with n variables and n 3 parameters introduced by gasper and rahman dev math 13 209 2005 generate a family of infinite dimensional modules for the q onsager algebra whose fundamental generators are realized in terms of the multivariable q difference and difference operators proposed by iliev trans am math soc 363 1577 2011 raising and lowering operators extending those of sahi sigma 3 002 2007 are also constructed in the second part finite dimensional modules are constructed and studied for a certain class of parameters and if the n variables belong to a discrete support in this case the bispectral property finds a natural interpretation within the framework of tridiagonal pairs in the third part eigenfunctions of the q dolan grady hierarchy are considered in the polynomial basis in particular invariant subspaces are identified for certain conditions generalizing nepomechie s relations in the fourth part the analysis is extended to the special case q 1 this framework provides a q hypergeometric formulation of quantum integrable models such as the open xxz spin chain with generic integrable boundary conditions q 1 for the class of quantum integrable models generated from the q onsager algebra a basis of bispectral multivariable q orthogonal polynomials is exhibited in the first part it is shown that the multivariable askey wilson polynomials with n variables and n 3 parameters introduced by gasper and rahman dev math 13 209 2005 generate a family of infinite dimensional modules for the q onsager algebra whose fundamental generators are realized in terms of the multivariable q difference and difference operators proposed by iliev trans am math soc 363 1577 2011 raising and lowering operators extending those of sahi sigma 3 002 2007 are also constructed in the second part finite dimensional modules are constructed and studied for a certain

the central extension of the reflection equations and an analog of miki s formula two different types of centrally extended quantum reflection algebras are introduced realizations in terms of the elements of the central extension of the yang baxter algebra are exhibited a co action map is identified for the special case of a realization in terms of elements satisfying the zamolodchikov faddeev algebra a boundary analog of miki s formula is also proposed providing a free field realization of q onsager currents

tridiagonal pairs of q racah type abstract let f denote an algebraically closed field and let v denote a vector space over f with finite positive dimension we consider a pair of linear transformations a v v and a v v that satisfy the following conditions i each of a a is diagonalizable ii there exists an ordering v i i 0 d of the eigenspaces of a such that a v i v i 1 v i v i 1 for 0 i d where v 1 0 and v d 1 0 iii there exists an ordering v i i 0 of the eigenspaces of a such that a v i v i 1 v i v i 1 for 0 i where v 1 0 and v 1 0 iv there is no subspace w of v such that a w w a w w w 0 w v we call such a pair a tridiagonal pair on v it is known that d for 0 i d let i resp i denote the eigenvalue of a resp a associated with v i resp v i the pair a a is said to have q racah type whenever i a b q 2 i d c q d 2 i and i a b q 2 i d c q d 2 i for 0 i d where q a b c a b c are scalars in f with q b c b c nonzero and q 2 1 1 this type is the most general one we classify up to isomorphism the tridiagonal pairs over f that have q racah type our proof involves the representation theory of the quantum affine algebra u q sl 2

modeling photonic crystals by boundary integral equations and dirichlet to neumann maps efficient numerical methods for analyzing photonic crystals phcs can be developed using the dirichlet to neumann dtn maps of the unit cells the dtn map is an operator that takes the wave field on the boundary of a unit cell to its normal derivative in frequency domain calculations for band structures and transmission spectra of finite phcs the dtn maps allow us to reduce the computation to the boundaries of the unit cells for two dimensional 2d phcs with unit cells containing circular cylinders the dtn maps can be constructed from analytic solutions the cylindrical waves in this paper we develop a boundary integral equation method for computing dtn maps of general unit cells containing cylinders with arbitrary cross sections the dtn map method is used to analyze band structures for 2d phcs with elliptic and other cylinders

form factors of twist fields in the lattice dirac theory we study u 1 twist fields in a two dimensional lattice theory of massive dirac fermions factorized formulas for finite lattice form factors of these fields are derived using elliptic parametrization of the spectral curve of the model elliptic determinant identities and theta functional interpolation we also investigate the thermodynamic and infinite volume scaling limit where the corresponding expressions reduce to form factors of the exponential fields of the sine gordon model at the free fermion point

form factors and complete spectrum of xxx antiperiodic higher spin chains by quantum separation of variables the antiperiodic transfer matrices associated to higher spin representations of the rational 6 vertex yang baxter algebra are analyzed by generalizing the approach introduced recently in the framework of sklyanin s quantum separation of variables sov for cyclic representations spin 1 2 highest weight representations and also for spin 1 2 representations of the 6 vertex reflection algebra such sov approach allow us to derive exactly results which represent complicate tasks for more traditional methods based on bethe ansatz and baxter q operator in particular we both prove the completeness of the sov characterization of the transfer matrix spectrum and its simplicity then the derived characterization of local operators by sklyanin s quantum separate variables and the expression of the scalar products of separate states by determinant formulae allow us to compute the form factors of the local spin operators by one determinant formulae similar to those of the scalar products

the riemann surface of the chiral potts model free energy function in a recent paper we derived the free energy or partition function of the n state chiral potts model by using the infinite lattice inversion relation method together with a non obvious extra symmetry this gave us three recursion relations for the partition function per site t pq of the infinite lattice here we use these recursion relations to obtain the full riemann surface of t pq in terms of the t p t q variables it consists of an infinite number of riemann sheets each sheet corresponding to a point on a 2n 1 dimensional lattice for n 2 the function t pq is meromorphic on this surface we obtain the orders of all the zeros and poles for n odd we show that these orders are determined by the usual inversion and rotation relations without the extra symmetry together with a simple linearity ansatz for n even this method does not give the orders uniquely but leaves only n 4 4 parameters to be determined

introduction to leonard pairs in this survey paper we give an elementary introduction to the theory of leonard pairs a leonard pair is defined as follows let k denote a field and let v denote a vector space over k with finite positive dimension by a leonard pair on v we mean an ordered pair of linear transformations a v v and b v v that satisfy conditions i ii below i there exists a basis for v with respect to which the matrix representing a is irreducible tridiagonal and the matrix representing b is diagonal ii there exists a basis for v with respect to which the matrix representing a is diagonal and the matrix representing b is irreducible tridiagonal we give several examples of leonard pairs using these we illustrate how leonard pairs arise in representation theory combinatorics and the theory of orthogonal polynomials

analysis of two dimensional photonic crystals using a multidomain pseudospectral method an analysis method based on a multidomain pseudospectral method is proposed for calculating the band diagrams of two dimensional photonic crystals and is shown to possess excellent numerical convergence behavior and accuracy the proposed scheme utilizes the multidomain chebyshev collocation method by applying chebyshev lagrange interpolating polynomials to the approximation of spatial derivatives at collocation points the helmholtz equation is converted into a matrix eigenvalue equation which is then solved for the eigenfrequencies by the shift inverse power method suitable multidomain division of the computational domain is performed to deal with general curved interfaces of the permittivity profile and field continuity conditions are carefully imposed across the dielectric interfaces the proposed method shows uniformly excellent convergence characteristics for both the transverse electric and transverse magnetic waves in the analysis of different structures the analysis of a mini band gap is also shown to demonstrate the extremely high accuracy of the proposed method

a lie algebra for closed strings spin chains and gauge theories we consider quantum dynamical systems whose degrees of freedom are described by n n matrices in the planar limit n examples are gauge theories and the m atrix theory of strings states invariant under u n are closed strings modeled by traces of products of matrices we have discovered that the u n invariant operators acting on both open and closed string states form a remarkable new lie algebra which we will call the heterix algebra the simplest special case with one degree of freedom is an extension of the virasoro algebra by the infinite dimensional general linear algebra furthermore these operators acting on closed string states only form a quotient algebra of the heterix algebra we will call this quotient algebra the cyclix algebra we express the hamiltonian of some gauge field theories like those with adjoint matter fields and dimensionally reduced pure qcd models as elements of this lie algebra finally we apply this cyclix algebra to establish an isomorphism between certain pl