rational and polynomial matrix factorizations via recursive pole zero cancellation we develop a recursive algorithm for obtaining factorizations of the type r r1 r2 where all three matrices are rational and r1 is nonsingular moreover the factors r1 and r2 are such that either the poles of r1 1 and r2 are in a prescribed region of the complex plane or their zeros such factorizations cover the specific cases of coprime factorization inner outer factorization gcd extraction and many more the algorithm works on the state space or generalized state space realization of r and derives in a recursive fashion the corresponding realizations of the factors

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

rank revealing qr factorizations abstract an algorithm is presented for computing a column permutation pi and a qr factorization a pi qr of an m by n m or n matrix a such that a possible rank deficiency of a will be revealed in the triangular factor r having a small lower right block for low rank deficient matrices the algorithm is guaranteed to reveal the rank of a and the cost is only slightly more than the cost of one regular qr factorization a posteriori upper and lower bounds on the singular values of a are derived and can be used to infer the numerical rank of a keywords qr factorization rank deficient matrices least squares computation subset selection rank singular values

eigenvalues invariant factors highest weights and schubert calculus we describe recent work of klyachko totaro knutson and tao that characterizes eigenvalues of sums of hermitian matrices and decomposition of tensor products of representations of gln c we explain related applications to invariant factors of products of matrices intersections in grassmann varieties and singular values of sums and products of arbitrary matrices recent breakthroughs primarily by a klyachko b totaro a knutson and t tao with contributions by p belkale and c woodward have led to complete solutions of several old problems involving the various notions in the title our aim here is to describe this work and especially to show how these solutions are derived from it along the way we will see that these problems are also related to other areas of mathematics including geometric invariant theory symplectic geometry and combinatorics in addition we present some related applications to singular values of arbitrary matrices although many of the theorems we state here have not appeared elsewhere their proofs are mostly soft algebra based on the hard geometric or combinatorial work of others indeed this paper emphasizes concrete elementary arguments we do give some new examples and counterexamples and raise some new open questions we have attempted to point to the sources and to some of the key partial results that had been conjectured or proved before however there is a very large literature particularly for linear algebra problems about eigenvalues singular values and invariant factors we have listed only a few of these articles from whose bibliographies we hope an interested reader can trace the history we apologize to the many whose work is not cited directly we begin in the first five sections by describing each of the problems together with some of their early histories and we state as theorems the new solutions to these problems in section 6 we describe the steps toward these solutions that were carried out before the recent breakthroughs then we discuss the recent solutions and explain how these theorems follow from the work of the above mathematicians sections 7 8 9 and 10 also contain variations and generalizations of some of the theorems stated in the first five sections as well as attributions of the theorems to their authors one of our fascinations with this subject even now that we have proofs of the theorems is the challenge to understand in a deeper way why all these subjects are received by the editors in july 1999 and in revised form january 3 2000 2000 mathematics subject classification primary 15a42 22e46 14m15 secondary 05e15 13f10 14c17 15a18 47b07 the author was partly supported by nsf grant dms9970435 c 2000 american mathematical society

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

an improved algorithm for the computation of structural invariants of a system pencil and related geometric aspects in this paper we propose a new recursive algorithm for computing the staircase form of a matrix pencil and implicitly its kronecker structure the algorithm compares favorably to existing ones in terms of elegance versatility and complexity in particular the algorithm without any modification yields the structural invariants associated with a generalized state space system and its system pencil two related geometric aspects are also discussed we show that an appropriate choice of a set of nested spaces related to the pencil leads directly to the staircase form we extend the notion of deflating subspace to the singular pencil case c 1997 elsevier science b v

an improved algorithm for the computation of kronecker s canonical form of a singular pencil we give an 0 t algorithm for computing the kronecker structure of an arbitrary m x n pencil xe a the algorithm is shown to he numerically stable because only unitary transformations are used the improved speed over earlier unitary methods is due to the efficient use of condensed forms which are maintained throughout the recursions of the algorithm

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

a simple state space design of an interactor for a non square system via system matrix pencil approach abstract the construction of an interactor cancelling the infinite zeros of a non square proper transfer matrix is discussed in this paper along the line of approach of b r copeland m g safonov int j robust nonlinear control 2 1992 139 in this paper some new properties of the infinite eigenstructure of the system matrix pencil of the transfer matrix are presented a simple state space design of the interactor is proposed by using these properties then some new features about state space relations among the original system the interactor and the compensated system are presented also several conditions about the invariance of the stabilizability or controllability between the state space realizations of the original system and the compensated system are shown furthermore the proposed method of designing the interactor in this paper can be applied to a tall or fat transfer matrix with full normal rank

a remark on the irreducible characters and fake degrees of finite real reflection groups beynon and lusztig have shown that the fake degrees of almost all irreducible characters of finite real reflection groups are palindromes and that the exceptions to this rule correspond to the non rational characters of the generic ringa defined overr c q their proof consists of a case by case check in this note we give an explanation for this phenomenon and some related facts about fake degrees moreover in the situation where we allow for distinct parametersq in the definition ofa we shall give a simple uniform proof of the fact that all the central idempotents of a bar k are elements of a tilde k where tilde k c sqrt q alpha

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

q krawtchouk polynomials as spherical functions on the hecke algebra of type b the hecke algebra for the hyperoctahedral group contains the hecke algebra for the symmetric group as a subalgebra inducing the index representation of the subalgebra gives a hecke algebra module which splits multiplicity free the corresponding zonal spherical functions are calculated in terms of q krawtchouk polynomials using the quantised enveloping algebra for sl 2 c the result covers a number of previously established interpretations of q krawtchouk polynomials on the hyperoctahedral group finite groups of lie type hypergroups and the quantum su 2 group

minimal indices cancellation and rank revealing factorizations for rational matrix functions we study the problem of eliminating the minimal indices the indices of a minimal polynomial basis of the null space of a rational matrix function by multiplication with a suitably chosen invertible rational matrix function we derive the class of all invertible factors that dislocate the minimal indices to certain zero locations and feature minimal mcmillan degree we impose additional conditions on the factor like being j unitary or j inner either with respect to the imaginary axis or to the unit circle and characterize the classes of solutions en route we extend the well known rank revealing factorization of a constant matrix to rational matrix functions the results are completely general and apply in particular to matrices which are polynomial strictly proper or improper rank deficient with arbitrary poles and zeros including at infinity all characterization are made by using descriptor realizations while the associated computations are performed by employing constant unitary transformations and standard reliable procedures for eigenvalue assignment or for solving riccati equations

zero cancellation for general rational matrix functions the problem of cancelling a specified part of the zeros of a completely general rational matrix function by multiplication with an appropriate invertible rational matrix function is investigated from different standpoints firstly the class of all factors that dislocate the zeros and feature minimal mcmillan degree are derived further necessary and sufficient existence conditions together with the construction of solutions are given when the factor fulfills additional assumptions like being j unitary or j inner either with respect to the imaginary axis or to the unit circle the main technical tool are centered realizations that deliver a sufficiently general conceptual support to cope with rational matrix functions which may be polynomial proper or improper rank deficient with arbitrary poles and zeros including at infinity a particular attention is paid to the numerically sound construction of solutions by employing at each stage unitary transformations reliable numerical algorithms for eigenvalue assignment and efficient lyapunov equation solvers

on a new class of realization formulas and their application abstract a new set of realization formulas is derived for a class of matrix valued functions w these include the standard realization formulas for rational w observability controllability and minimality are defined and characterized conditions for w to be j 1 j 2 isometric and or j 1 j 2 coisometric with respect to a pair of signature matrices j 1 and j 2 are given in terms of the realizations minimal factorizations for square w are considered and formulas for the factors are deduced associated reproducing kernel spaces of pairs too are discussed

the generalized eigenstructure problem in linear system theory the algebraic theory of linear time invariant systems has been studied in large detail during the past few decades and numerous computational algorithms have been developed to solve problems arising in this context in this paper the numerical aspects of a certain class of such algorithms dealing with what the author calls generalized eigenstructure problems are discussed some new and or modified algorithms are presented both the nmnerical stability of the algorithms and the conditioning of the problems they solve are analyzed using numerical criteria

two linear transformations each tridiagonal with respect to an eigenbasis of the other the td d canonical form and the lb ub canonical form abstract let k denote a field and let v denote a vector space over k with finite positive dimension we consider an ordered pair of linear transformations a v v and b v v which satisfy both 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 call such a pair a leonard pair on v we introduce two canonical forms for leonard pairs we call these the td d canonical form and the lb ub canonical form in the td d canonical form the leonard pair is represented by an irreducible tridiagonal matrix and a diagonal matrix subject to a certain normalization in the lb ub canonical form the leonard pair is represented by a lower bidiagonal matrix and an upper bidiagonal matrix subject to a certain normalization we describe the two canonical forms in detail as an application we obtain the following results given square matrices a b over k with a tridiagonal and b diagonal we display a necessary and sufficient condition for a b to represent a leonard pair given square matrices a b over k with a lower bidiagonal and b upper bidiagonal we display a necessary and sufficient condition for a b to represent a leonard pair we briefly discuss how leonard pairs correspond to the q racah polynomials and some related polynomials in the askey scheme we present some open problems concerning leonard pairs

the quantum algebra uq su2 and q krawtchouk families of polynomials the aim of this paper is to study quantum and affine q krawtchouk polynomials by means of operators of irreducible representations of the quantum algebra uq su2 we diagonalize a certain operator i in such a representation and show that elements of the transition matrix from the initial canonical basis to the basis consisting of eigenfunctions of the operator i are expressed in terms of quantum q krawtchouk polynomials then we find an explicit form of the operator in the basis of the eigenfunctions of i in which it has the form of a jacobi matrix normalizing this basis and using the operator we thus derive the orthogonality relations for quantum and affine q krawtchouk polynomials we show that affine q krawtchouk polynomials are dual to quantum q krawtchouk polynomials a biorthogonal system of functions with respect to the scalar product in the representation space is also derived

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

conformal invariance and critical spectrum of corner transfer matrices the finite size scaling spectra of ising model corner transfer matrices and their generators are studied at criticality the generators are diagonalised using fermion algebra the low lying eigenvalues given by the zeros of meixner polynomials are equally spaced and collapse like 1 log n for large n as predicted by conformal invariance the asymptotics are obtained using a generalised euler maclaurin summation formula the shift in the largest eigenvalue is given analytically as pi c 6 log n with central charge c 1 2 the spectrum generating functions for both fixed and free boundary conditions are expressed simply in terms of the c 1 2 virasoro characters chi delta q with modular parameter q exp pi log n and conformal dimensions delta 0 1 2 1 16

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

mehler heine type formulas for the krawtchouk polynomials abstract we derive mehler heine type asymptotic expansions for the krawtchouk polynomials k n x p n with n o n and x o n these formulas provide good approximations for k n x p n and determine the asymptotic limit of their zeros as n