towers of recollement and bases for diagram algebras planar diagrams and a little beyond abstract the recollement approach to the representation theory of sequences of algebras is extended to pass basis information directly through the globalisation functor the method is hence adapted to treat sequences that are not necessarily towers by inclusion such as symplectic blob algebras diagram algebra quotients of the type c hecke algebras by carefully reviewing the diagram algebra construction we find a new set of functors interrelating module categories of ordinary blob algebras diagram algebra quotients of the type b hecke algebras at different values of the algebra parameters we show that these functors generalise to determine the structure of symplectic blob algebras and hence of certain two boundary temperley lieb algebras arising in statistical mechanics we identify the diagram basis with a cellular basis for each symplectic blob algebra and prove that these algebras are quasihereditary over a field for almost all parameter choices and generically semisimple that is we give bases for all cell and standard modules

uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory we consider asymptotics for orthogonal polynomials with respect to varying exponential weights wn x dx e nv x dx on the line as n the potentials v are assumed to be real analytic with sufficient growth at infinity the principle results concern plancherel rotach type asymptotics for the orthogonal polynomials down to the axis using these asymptotics we then prove universality for a variety of statistical quantities arising in the theory of random matrix models some of which have been considered recently in 31 and also in 4 an additional application concerns the asymptotics of the recurrence coefficients and leading coefficients for the orthonormal polynomials see also 4 the orthogonal polynomial problem is formulated as a riemann hilbert problem following 19 20 the riemann hilbert problem is analyzed in turn using the steepest descent method introduced in 12 and further developed in 11 13 a critical role in our method is played by the equilibrium measure d v for v as analyzed in 8 1999 john wiley sons inc

correlation functions cluster functions and spacing distributions for random matrices the usual formulas for the correlation functions in orthogonal and symplectic matrix models express them as quaternion determinants from this representation one can deduce formulas for spacing probabilities in terms of fredholm determinants of matrix valued kernels the derivations of the various formulas are somewhat involved in this article we present a direct approach which leads immediately to scalar kernels for the unitary ensembles and matrix kernels for the orthogonal and symplectic ensembles and the representations of the correlation functions cluster functions and spacing distributions in terms of them

the τ function of the ablowitz segur family of solutions to painleve ii as a widom constant the functions of certain painleve equations pvi pv and piii can be expressed as fredholm determinants furthermore the minor expansion of these determinants provides an interesting connection to random partitions this paper is a step toward understanding whether the function of painleve ii has a fredholm determinant representation the ablowitz segur family of solutions are special one parameter solutions of painleve ii and the corresponding function is known to be the fredholm determinant of the airy kernel we develop a formalism for open contour in parallel to the one formulated in terms of a suitable combination of toeplitz operators called the widom constant and verify that the widom constant for the ablowitz segur family of solutions is indeed the determinant of the airy kernel finally we construct a suitable basis and obtain the minor expansion of the ablowitz segur function the functions of certain painleve equations pvi pv and piii can be expressed as fredholm determinants furthermore the minor expansion of these determinants provides an interesting connection to random partitions this paper is a step toward understanding whether the function of painleve ii has a fredholm determinant representation the ablowitz segur family of solutions are special one parameter solutions of painleve ii and the corresponding function is known to be the fredholm determinant of the airy kernel we develop a formalism for open contour in parallel to the one formulated in terms of a suitable combination of toeplitz operators called the widom constant and verify that the widom constant for the ablowitz segur family of solutions is indeed the determinant of the airy kernel finally we construct a suitable basis and obtain the minor expansion of the ablowitz segur function

asymptotics of plancherel measures for the infinite dimensional unitary group we study a two dimensional family of probability measures on infinite gelfand tsetlin schemes induced by a distinguished family of extreme characters of the infinite dimensional unitary group these measures are unitary group analogs of the well known plancherel measures for symmetric groups we show that any measure from our family defines a determinantal point process on z x z and we prove that in appropriate scaling limits such processes converge to two different extensions of the discrete sine process as well as to the extended airy and pearcey processes

averages of characteristic polynomials in random matrix theory we compute averages of products and ratios of characteristic polynomials associated with orthogonal unitary and symplectic ensembles of random matrix theory the pfaffian determinantal formulae for these averages are obtained and the bulk scaling asymptotic limits are found for ensembles with gaussian weights classical results for the correlation functions of the random matrix ensembles and their bulk scaling limits are deduced from these formulae by a simple computation we employ a discrete approximation method the problem is solved for discrete analogues of random matrix ensembles originating from representation theory and then a limit transition is performed exact pfaffian determinantal formulae for the discrete averages are proven using standard tools of linear algebra no application of orthogonal or skew orthogonal polynomials is needed 2005 wiley periodicals inc

z measures on partitions related to the infinite gelfand pair s 2 h abstract the paper deals with the z measures on partitions with the deformation jack parameters 2 or 1 2 we provide a detailed explanation of the representation theoretic origin of these measures and of their role in the harmonic analysis on the infinite symmetric group

the z measures on partitions pfaffian point processes and the matrix hypergeometric kernel abstract we consider a point process on one dimensional lattice originated from the harmonic analysis on the infinite symmetric group and defined by the z measures with the deformation jack parameter 2 we derive an exact pfaffian formula for the correlation function of this process namely we prove that the correlation function is given as a pfaffian with a 2 2 matrix kernel the kernel is given in terms of the gauss hypergeometric functions and can be considered as a matrix analogue of the hypergeometric kernel introduced by a borodin and g olshanski 2000 5 our result holds for all values of admissible complex parameters

the partition algebra and the potts model transfer matrix spectrum in high dimensions we construct generalizations pmn q of the partition algebra pn q martin p p 1996 j algebra 183 319 facilitating a representation theoretic approach to the n site transfer matrix spectrum of a high dimensional q state potts model with magnetic field and source terms and to corresponding dichromatic polynomials for each q we describe the irreducible representation theory of the sequence of algebras p q pn q pn1 q pn 1 q n 0 1 2 approaching the large n limit for each positive integer q we extend the potts model representation n of pn q to a representation of p1n q we show how these potts representations embed in the representation theory of the partition algebras these results together provide a tool with which to examine the nature of physical correlation functions for large n the irreducible content of the potts representations can be summarized by sq endpn q vq n and sq 1 endpn1 q vq n where sq is the symmetric group and vq is the space of states of a potts spin we show how the partition algebra formalism matches up the correlation functions of the potts model and the corresponding absolute spectrum degeneracies of its transfer matrix

bc type z measures and determinantal point processes abstract the bc type z measures are a family of four parameter z z a b probability measures on the path space of the nonnegative gelfand tsetlin graph with jacobi edge multiplicities we can interpret the z measures as random point processes p z z a b on the punctured positive real line x r 0 1 our main result is that these random processes are determinantal and moreover we compute their correlation kernels explicitly in terms of hypergeometric functions for very special values of the parameters z z the processes p z z a b on x are essentially scaling limits of racah orthogonal polynomial ensembles and their correlation kernels can be computed simply from some limits of the racah polynomials thus in the language of random matrices we study certain analytic continuations of processes that are limits of racah ensembles and such that they retain the determinantal structure another interpretation of our results and the main motivation of this paper is the representation theory of big groups in representation theoretic terms this paper solves a natural problem of harmonic analysis for several infinite dimensional symmetric spaces

ramified partition algebras for each natural number n poset t and t tuple of scalars q we introduce the ramified partition algebra q which is a physically motivated and natural generalization of the partition algebra 24 25 the partition algebra coincides with case t 1 for fixed n and t these algebras like the partition algebra have a basis independent of q we investigate their representation theory in case t 1 2 leq we show that q is quasi hereditary over field k when q 1 q 2 is invertible in k and k is such that certain finite group algebras over k are semisimple e g when k is algebraically closed characteristic zero under these conditions we determine an index set for simple modules of q and construct standard modules with this index set we show that there are unboundedly many choices of q such that q is not semisimple for sufficiently large n but that it is generically semisimple for all n we construct tensor space representations of certain non semisimple specializations of q and show how to use these to build clock model transfer matrices 24 in arbitrary physical dimensions

determinantal point processes with j hermitian correlation kernels let x be a locally compact polish space and let m be a reference radon measure on x let x denote the configuration space over x that is the space of all locally finite subsets of x a point process on x is a probability measure on x a point process is called determinantal if its correlation functions have the form k n x1 xn det k xi xj i j 1 n the function k x y is called the correlation kernel of the determinantal point process assume that the space x is split into two parts x x1 x2 a kernel k x y is called j hermitian if it is hermitian on x1 x1 and x2 x2 and k x y k y x for x x1 and y x2 we derive a necessary and sufficient condition of existence of a determinantal point process with a j hermitian correlation kernel k x y

correlation kernels for discrete symplectic and orthogonal ensembles in 49 h widom derived formulae expressing correlation functions of orthogonal and symplectic ensembles of random matrices in terms of orthogonal polynomials we obtain similar results for discrete ensembles with rational discrete logarithmic derivative and compute explicitly correlation kernels associated to the classical meixner and charlier orthogonal polynomials

distribution of the first particle in discrete orthogonal polynomial ensembles we show that the distribution function of the first particle in a discrete orthogonal polynomial ensemble can be obtained through a certain recurrence procedure if the difference or q log derivative of the weight function is rational in a number of classical special cases the recurrence procedure is equivalent to the difference and q painleve equations of 10 17 our approach is based on the formalism of discrete integrable operators and discrete riemann hilbert problems developed in 3 4

integrals over grassmannians and random permutations testing the independence of two gaussian populations involves the distribution of the sample canonical correlation coefficients given that the actual correlation is zero the laplace transform as a function of x of this distribution is not only an integral over the grassmannian gr p f n of p dimensional planes in real complex or quaternion n space f n but is also related to a generalized hypergeometric function such integrals are solutions of painleve like equations in the complex case they are solutions to genuine painleve equations these integrals over gr p c n have remarkable expansions in x related to random words of length l formed with an alphabet of p letters 1 p the coefficients of these expansions are given by the probability that a word i contains a subsequence of letters p p 1 1 in that order and ii that the maximal length of the disjoint union of p 1 increasing subsequences of the word is less than or equal tok where k refers to the power of x note that if each letter appears in the word then the maximal length of the disjoint union of p increasing subsequences of the word is automatically l and is thus trivial c 2003 elsevier inc all rights reserved

phase transitions and equilibrium measures in random matrix models the paper is devoted to a study of phase transitions in the hermitian random matrix models with a polynomial potential in an alternative equivalent language we study families of equilibrium measures on the real line in a polynomial external field the total mass of the measure is considered as the main parameter which may be interpreted also either as temperature or time our main tools are differentiation formulas with respect to the parameters of the problem and a representation of the equilibrium potential in terms of a hyperelliptic integral using this combination we introduce and investigate a dynamical system system of odes describing the evolution of families of equilibrium measures on this basis we are able to systematically derive a number of new results on phase transitions such as the local behavior of the system at all kinds of phase transitions as well as to review a number of known ones

janossy densities ii pfaffian ensembles we extend the main result of the companion paper j stat phys 113 595 610 to the case of the pfaffian ensembles

on representations of fuss catalan algebras abstract in this paper we study the representation theory of the fuss catalan algebras fc n a b we prove that this algebra is cellular with a cellular basis and forms a tower of recollement as defined by cox martin parker and xi 7 and hence it is quasi hereditary algebra if a b are non zero complex numbers

mixed schur weyl sergeev duality for queer lie superalgebras abstract we introduce a new family of superalgebras b r s for r s 0 such that r s 0 which we call the walled brauer superalgebras and prove the mixed schur weyl sergeev duality for queer lie superalgebras more precisely let q n be the queer lie superalgebra v c n n the natural representation of q n and w the dual of v we prove that if n r s the superalgebra b r s is isomorphic to the supercentralizer algebra end q n v r w s op of the q n action on the mixed tensor space v r w s as an ingredient for the proof of our main result we construct a new diagrammatic realization d k of the sergeev superalgebra ser k finally we give a presentation of b r s in terms of generators and relations

shape fluctuations and random matrices we study a certain random growth model in two dimensions closely related to the one dimensional totally asymmetric exclusion process the results show that the shape fluctuations appropriately scaled converges in distribution to the tracy widom largest eigenvalue distribution for the gaussian unitary ensemble gue

markov processes on partitions we introduce and study a family of markov processes on partitions the processes preserve the so called z measures on partitions previously studied in connection with harmonic analysis on the infinite symmetric group we show that the dynamical correlation functions of these processes have determinantal structure and we explicitly compute their correlation kernels we also compute the scaling limits of the kernels in two different regimes the limit kernels describe the asymptotic behavior of large rows and columns of the corresponding random young diagrams and the behavior of the young diagrams near the diagonal

interpretations of some parameter dependent generalizations of classical matrix ensembles two types of parameter dependent generalizations of classical matrix ensembles are defined by their probability density functions pdfs as the parameter is varied one interpolates between the eigenvalue pdf for the superposition of two classical ensembles with orthogonal symmetry and the eigenvalue pdf for a single classical ensemble with unitary symmetry while the other interpolates between a classical ensemble with orthogonal symmetry and a classical ensemble with symplectic symmetry we give interpretations of these pdfs in terms of probabilities associated to the continuous robinson schensted knuth correspondence between matrices with entries chosen from certain exponential distributions and non intersecting lattice paths and in the course of this probability measures on partitions and pairs of partitions are identified the latter are generalized by using macdonald polynomial theory and a particular continuum limit the jacobi limit of the resulting measures is shown to give pdfs related to those appearing in the work of anderson on the selberg integral and also in some classical work of dixon by interpreting anderson s and dixon s work as giving the pdf for the zeros of a certain rational function it is then possible to identify random matrices whose eigenvalue pdfs realize the original parameter dependent pdfs this line of theory allows sampling of the original parameter dependent pdfs their dixon anderson type generalizations and associated marginal distributions from the zeros of certain polynomials defined in terms of random three term recurrences

janossy densities i determinantal ensembles we derive an elementary formula for janossy densities for determinantal point processes with a finite rank projection type kernel in particular for 2 polynomial ensembles of random matrices we show that the janossy densities on an interval i can be expressed in terms of the christoffel darboux kernel for the orthogonal polynomials on the complement of i

star triangle relation for a three dimensional model the solvablesl n chiral potts model can be interpreted as a three dimensional lattice model with local interactions to within a minor modification of the boundary conditions it is an ising type model on the body centered cubic lattice with two and three spin interactions the corresponding local boltzmann weights obey a number of simple relations including a restricted star triangle relation which is a modified version of the well known star triangle relation appearing in two dimensional models we show that these relations lead to remarkable symmetry properties of the boltzmann weight function of an elementary cube of the lattice related to the spatial symmetry group of the cubic lattice these symmetry properties allow one to prove the commutativity of the row to row transfer matrices bypassing the tetrahedron relation the partition function per site for the infinite lattice is calculated exactly

two dimensional yang mills theory painleve equations and the six vertex model we show that the chiral partition function of two dimensional yang mills theory on the sphere can be mapped to the partition function of the homogeneous six vertex model with domain wall boundary conditions in the ferroelectric phase a discrete matrix model description in both cases is given by the meixner ensemble leading to a representation in terms of a stochastic growth model we show that the partition function is a particular case of the z measure on the set of young diagrams yielding a unitary matrix model for chiral yang mills theory on s2 and the identification of the partition function as a tau function of the painlev v equation we describe the role played by generalized non chiral yang mills theory on s2 in relating the meixner matrix model to the toda chain hierarchy encompassing the integrability of the six vertex model we also argue that the thermodynamic behaviour of the six vertex model in the disordered and antiferroelectric phases are captured by particular q deformations of two dimensional yang mills theory on the sphere

janossy densities of coupled random matrices we explicitly calculate the janossy densities for a special class of finite determinantal random point processes with several types of particles introduced by prahofer and spohn and in the full generality by johansson in connection with the analysis of polynuclear growth models the results of this paper generalize the theorem we proved earlier with borodin about the janossy densities in biorthogonal ensembles in particular our results can be applied to ensembles of random matrices coupled in a chain which provide a very important example of determinantal ensembles we study

the boundary of the gelfand tsetlin graph a new approach the gelfand tsetlin graph is an infinite graded graph that encodes branching of irreducible characters of the unitary groups the boundary of the gelfand tsetlin graph has at least three incarnations as a discrete potential theory boundary as the set of finite indecomposable characters of the infinite dimensional unitary group and as the set of doubly infinite totally positive sequences an old deep result due to albert edrei and dan voiculescu provides an explicit description of the boundary it can be realized as a region in an infinite dimensional coordinate space the paper contains a novel approach to the edrei voiculescu theorem it is based on a new explicit formula for the number of semi standard young tableaux of a given skew shape or of gelfand tsetlin schemes of trapezoidal shape the formula is obtained via the theory of symmetric functions and new schur like symmetric functions play a key role in the derivation

kernels and point processes associated with whittaker functions this article considers whittaker s confluent hypergeometric function w where is real and is real or purely imaginary then x x 1 2w x arises as the scattering function of a continuous time linear system with state space l2 1 2 and input and output spaces c the hankel operator on l2 0 is expressed as a matrix with respect to the laguerre basis and gives the hankel matrix of moments of a jacobi weight w0 x xb 1 x a the operation of translating is equivalent to deforming w0 to give wt x e t xxb 1 x a the determinant of the hankel matrix of moments of we satisfies the form of painleve s transcendental differential equation pv it is shown that gives rise to the whittaker kernel from random matrix theory as studied by borodin and olshanski commun math phys 211 335 358 2000 whittaker kernels are closely related to systems of orthogonal polynomials for a pollaczek jacobi type weight lying outside the usual szego class

approximate solution of nonlinear double scattering inversion for true amplitude imaging abstractlinearized inversion algorithms are the main techniques in seismic imaging that apply the single scattering born approximation to the scattered field and therefore have difficulty handling the strong perturbation of model parameters and nonlinear multiple scattering effects to theoretically overcome these drawbacks in the linearization of the inverse scattering problem we have developed an approach to deal with nonlinear double scattering inversion we first used an integral equation formulation associated with the scattered field consisting of single and double scattering in an acoustic earth model based on the second order born approximation and we found that the approximation of the scattered field can be naturally related to the generalized radon transform grt we then adopted the inverse grt to obtain the corresponding quadratic approximate inversion solution our inversion scheme can appropriately handle the first order transmission effects from double scattering in a local area wh

the sensitivity of born inversion to the choice of reference velocity a simple example we examine the sensitivity of the born model to the input background velocity we use a one dimensional analytic example to point out the difference between a corrective procedure and merely a perturbative one we examine various aspects of the sensitivity issue including the trade off between velocity determination and mapping of reflector location although this problem is discussed within the context of the born model it is an issue common to all perturbative methods e g migration methods which transform surface reflection data into a map of subsurface reflectors

harmonic analysis in phase space and finite weyl heisenberg ensembles weyl heisenberg ensembles are translation invariant determinantal point processes on mathbb r 2d associated with the schrodinger representation of the heisenberg group and include as examples the ginibre ensemble and the polyanalytic ensembles which model the higher landau levels in physics we introduce finite versions of the weyl heisenberg ensembles and show that they behave analogously to the finite ginibre ensembles more specifically guided by the observation that the ginibre ensemble with n points is asymptotically close to the restriction of the infinite ginibre ensemble to the disk of area n we define finite wh ensembles as adequate finite approximations of the restriction of infinite wh ensembles to a given domain omega we provide a precise rate for the convergence of the corresponding one point intensities to the indicator function of omega as omega is dilated and the process is rescaled proportionally thermodynamic regime the construction and analysis rely neither on explicit formulas nor on the asymptotics for orthogonal polynomials but rather on phase space methods second we apply our construction to study the pure finite ginibre type polyanalytic ensembles which model finite particle systems in a single landau level and are defined in terms of complex hermite polynomials on a technical level we show that finite wh ensembles provide an approximate model for finite polyanalytic ginibre ensembles and we quantify the corresponding deviation by means of this asymptotic description we derive estimates for the rate of convergence of the one point intensity of polyanalytic ginibre ensembles in the thermodynamic limit

eigenvalue statistics of the real ginibre ensemble the real ginibre ensemble consists of random n ifmmode times else texttimes fi n matrices formed from independent and identically distributed standard gaussian entries by using the method of skew orthogonal polynomials the general n point correlations for the real eigenvalues and for the complex eigenvalues are given as n ifmmode times else texttimes fi n pfaffians with explicit entries a computationally tractable formula for the cumulative probability density of the largest real eigenvalue is presented this is relevant to may s stability analysis of biological webs

on an algebraic approach to higher dimensional statistical mechanics we study representations of temperley lieb algebras associated with the transfer matrix formulation of statistical mechanics on arbitrary lattices we first discuss a new hyperfinite algebra the diagram algebra d underset raise0 3em hbox smash scriptscriptstyle n q which is a quotient of the temperley lieb algebra appropriate for potts models in the mean field case and in which the algebras appropriate for all transverse lattice shapesg appear as subalgebras we give the complete structure of this subalgebra in the case hat a n potts model on a cylinder the study of the full temperley lieb algebra of graphg reveals a vast number of infinite sets of inequivalent irreducible representations characterized by one or more complex parameters associated to topological effects such as links we give a complete classification in the hat a n case where the only such effects are loops and twists

infinite dimensional diffusions as limits of random walks on partitions starting with finite markov chains on partitions of a natural number n we construct via a scaling limit transition as n a family of infinite dimensional diffusion processes the limit processes are ergodic their stationary distributions the so called z measures appeared earlier in the problem of harmonic analysis for the infinite symmetric group the generators of the processes are explicitly described

harmonic analysis on the infinite symmetric group the infinite symmetric group s whose elements are finite permutations of 1 2 3 is a model example of a big group by virtue of an old result of murray von neumann the one sided regular representation of s in the hilbert space l2 s generates a type ii1 von neumann factor while the two sided regular representation is irreducible this shows that the conventional scheme of harmonic analysis is not applicable to s for the former representation decomposition into irreducibles is highly non unique and for the latter representation there is no need of any decomposition at all we start with constructing a compactification mathfrak s supset s infty which we call the space of virtual permutations although mathfrak s is no longer a group it still admits a natural two sided action of s thus mathfrak s is a g space where g stands for the product of two copies of s on mathfrak s there exists a unique g invariant probability measure 1 which has to be viewed as a true haar measure for s more generally we include 1 into a family t t 0 of distinguished g quasiinvariant probability measures on virtual permutations by making use of these measures we construct a family tz z of unitary representations of g called generalized regular representations each representation tz with z 0 can be realized in the hilbert space l 2 mathfrak s mu t where t z 2 as z the generalized regular representations tz approach in a suitable sense the naive two sided regular representation of the group g in the space l2 s in contrast with the latter representation the generalized regular representations tz are highly reducible and have a rich structure we prove that any tz admits a unique decomposition into a multiplicity free continuous integral of irreducible representations of g for any two distinct and not conjugate complex numbers z1 z2 the spectral types of the representations t z 1 and t z 2 are shown to be disjoint in the case z a complete description of the spectral type is obtained further work on the case z reveals a remarkable link with stochastic point processes and random matrix theory

on representation categories of wreath products in non integral rank abstract for an arbitrary commutative ring k and t k we construct a 2 functor s t which sends a tensor category to a new tensor category by applying it to the representation category of a bialgebra we obtain a family of categories which interpolates the representation categories of the wreath products of the bialgebra this generalizes the construction of deligne s category rep s t k for representation categories of symmetric groups

derivation of forward and adjoint operators for least squares shot profile split step migration the forward and adjoint operators for shot profile least squares migration are derived the forward operator is demigration and the adjoint operator is migration the demigration operator is derived from the born approximation the process begins with a green s function that allows for a laterally varying migration velocity model using the split step approximation next the earth is divided into horizontal layers and within each layer the migration velocity model is made to be constant with respect to depth for a given layer 1 the source side wavefield is propagated down to its top using the background wavefield this gives a background wavefield incident at the layer s upper boundary 2 the layer s contribution to the scattered wavefield is computed using the born approximation to the scattered wavefield and the background wavefield 3 next its scattered wavefield is propagated back up to the measurement surface using again the background wavefield the measured wavefield is approximated by the sum of scattered wavefields from each layer in the derivation of the measured wavefield the shot profile migration geometry is used for each shot the resulting wavefield modeling operator takes the form of a fredholm integral equation of the first kind and this is used to write down its adjoint the shot profile migration operator this forward adjoint pair is used for shot profile least squares migration shot profile least squares migration is illustrated with two synthetic examples the first uses data collected over a four layer acoustic model and the second uses data from the sigsbee 2a model

on linear systems and τ functions associated with lame s equation and painleve s equation vi abstract painleve s transcendental differential equation p vi may be expressed as the consistency condition for a pair of linear differential equations with 2 2 matrix coefficients with rational entries by a construction due to tracy and widom this linear system is associated with certain kernels which give trace class operators on hilbert space this paper expresses such operators in terms of hankel operators of linear systems which are realised in terms of the laurent coefficients of the solutions of the differential equations let p t l 2 0 l 2 t be the orthogonal projection then the fredholm determinant t det i p t defines the function which is here expressed in terms of the solution of a matrix gelfand levitan equation for suitable values of the parameters solutions of the hypergeometric equation give a linear system with similar properties for meromorphic transfer functions that have poles on an arithmetic progression the corresponding hankel operator has a simple form with respect to an exponential basis in l 2 0 so det i p t can be expressed as a series of finite determinants this applies to elliptic functions of the second kind such as satisfy lame s equation with l 1

j hermitian determinantal point processes balanced rigidity and balanced palm equivalence we study palm measures of determinantal point processes with j hermitian correlation kernels a point process mathbb p on the punctured real line mathbb r mathbb r sqcup mathbb r is said to be balanced rigid if for any precompact subset b subset mathbb r the difference between the numbers of particles of a configuration inside b cap mathbb r and b cap mathbb r is almost surely determined by the configuration outside b the point process mathbb p is said to have the balanced palm equivalence property if any reduced palm measure conditioned at 2n distinct points n in mathbb r and n in mathbb r is equivalent to mathbb p we formulate general criteria for determinantal point processes with j hermitian correlation kernels to be balanced rigid and to have the balanced palm equivalence property and prove in particular that the determinantal point processes with whit taker kernels of borodin and olshanski are balanced rigid and have the balanced palm equivalence property

correlations for superpositions and decimations of laguerre and jacobi orthogonal matrix ensembles with a parameter a superposition of a matrix ensemble refers to the ensemble constructed from two independent copies of the original while a decimation refers to the formation of a new ensemble by observing only every second eigenvalue in the cases of the classical matrix ensembles with orthogonal symmetry it is known that forming superpositions and decimations gives rise to classical matrix ensembles with unitary and symplectic symmetry the basic identities expressing these facts can be extended to include a parameter which in turn provides us with probability density functions which we take as the definition of special parameter dependent matrix ensembles the parameter dependent ensembles relating to superpositions interpolate between superimposed orthogonal ensembles and a unitary ensemble while the parameter dependent ensembles relating to decimations interpolate between an orthogonal ensemble with an even number of eigenvalues and a symplectic ensemble of half the number of eigenvalues by the construction of new families of biorthogonal and skew orthogonal polynomials we are able to compute the corresponding correlation functions both in the finite system and in various scaled limits specializing back to the cases of orthogonal and symplectic symmetry we find that our results imply different functional forms to those known previously

harmonic analysis on the infinite dimensional unitary group and determinantal point processes the infinite dimensional unitary group u is the inductive limit of growing compact unitary groups u n in this paper we solve a problem of harmonic analysis on u stated in ol3 the problem consists in computing spectral decomposition for a remarkable 4 parameter family of characters of u these characters generate representations which should be viewed as analogs of nonexisting regular representation of u the spectral decomposition of a character of u is described by the spectral measure which lives on an infinite dimensional space of indecomposable characters the key idea which allows us to solve the problem is to embed into the space of point configurations on the real line without two points this turns the spectral measure into a stochastic point process on the real line the main result of the paper is a complete description of the processes corresponding to our concrete family of characters we prove that each of the processes is a determinantal point process that is its correlation functions have determinantal form with a certain kernel our kernels have a special integrable form and are expressed through the gauss hypergeometric function from the analytic point of view the problem of computing the correlation kernels can be reduced to a problem of evaluating uniform asymptotics of certain discrete orthogonal polynomials studied earlier by richard askey and peter lesky one difficulty lies in the fact that we need to compute the asymptotics in the oscillatory regime with the period of oscillations tending to 0 we do this by expressing the polynomials in terms of a solution of a discrete riemann hilbert problem and computing the nonoscillatory asymptotics of this solution from the point of view of statistical physics we study thermodynamic limit of a discrete log gas system an interesting feature of this log gas is that its density function is asymptotically equal to the characteristic function of an interval our point processes describe how different the random particle configuration is from the typical densely packed configuration in simpler situations of harmonic analysis on infinite symmetric groups and harmonic analysis of unitarily invariant measures on infinite hermitian matrices similar results were obtained in our papers bo1 bo2 bo4

fredholm determinants jimbo miwa ueno τ functions and representation theory the authors show that a wide class of fredholm determinants arising in the representation theory of big groups such as the infinite dimensional unitary group solve painleve equations their methods are based on the theory of integrable operators and the theory of riemann hilbert problems 2002 wiley periodicals inc

duality of orthogonal polynomials on a finite set we prove a certain duality relation for orthogonal polynomials defined on a finite set the result is used in a direct proof of the equivalence of two different ways using particles or holes of computing the correlation functions of a discrete orthogonal polynomial ensemble

matrix kernels for measures on partitions we consider the problem of computation of the correlation functions for the z measures with the deformation jack parameters 2 or 1 2 such measures on partitions are originated from the representation theory of the infinite symmetric group and in many ways are similar to the ensembles of random matrix theory of 4 or 1 symmetry types for a certain class of such measures we show that correlation functions can be represented as pfaffians including 2 2 matrix valued kernels and compute these kernels explicitly we also give contour integral representations for correlation kernels of closely connected measures on partitions

α pfaffian pfaffian point process and shifted schur measure abstract for any complex number and any even size skew symmetric matrix b we define a generalization pf b of the pfaffian pf b which we call the pfaffian the pfaffian is a pfaffian analogue of the determinant studied in t shirai and y takahashi j funct anal 205 2003 414 463 and d vere jones linear algebra appl 111 1988 119 124 it gives the pfaffian at 1 we give some formulas for pfaffians and study the positivity further we define point processes determined by the pfaffian also we provide a linear algebraic proof of the explicit pfaffian expression obtained in s matsumoto correlation functions of the shifted schur measure math co 0312373 for the correlation function of the shifted schur measure

discrete orthogonal polynomial ensembles and the plancherel measure we consider discrete orthogonal polynomial ensembles which are discrete analogues of the orthogonal polynomial ensembles in random matrix theory these ensembles occur in certain problems in combinatorial probability and can be thought of as probability measures on partitions the meixner ensemble is related to a two dimensional directed growth model and the charlier ensemble is related to the lengths of weakly increasing subsequences in random words the krawtchouk ensemble occurs in connection with zig zag paths in random domino tilings of the aztec diamond and also in a certain simplifled directed flrst passage percolation model we use the charlier ensemble to investigate the asymptotics of weakly increasing subsequences in random words and to prove a conjecture of tracy and widom as a limit of the meixner ensemble or the charlier ensemble we obtain the plancherel measure on partitions and using this we prove a conjecture of baik deift and johansson that under the plancherel measure the distribution of the lengths of the flrst k rows in the partition appropriately scaled converges to the asymptotic joint distribution for the k largest eigenvalues of a random matrix from the gaussian unitary ensemble in this problem a certain discrete kernel which we call the discrete bessel kernel plays an important role

distributions on partitions point processes and the hypergeometric kernel we study a 3 parametric family of stochastic point processes on the one dimensional lattice originated from a remarkable family of representations of the infinite symmetric group we prove that the correlation functions of the processes are given by determinantal formulas with a certain kernel the kernel can be expressed through the gauss hypergeometric function we call it the hypergeometric kernel in a scaling limit our processes approximate the processes describing the decomposition of representations mentioned above into irreducibles as we showed in previous works the correlation functions of these limit processes also have determinantal form with so called whittaker kernel we show that the scaling limit of the hypergeometric kernel is the whittaker kernel integrable operator as defined by its izergin korepin and slavnov we argue that the hypergeometric kernel can be considered as a kernel defining a discrete integrable operator we also show that the hypergeometric kernel degenerates for certain values of parameters to the christoffel darboux kernel for meixner orthogonal polynomials this fact is parallel to the degeneration of the whittaker kernel to the christoffel darboux kernel for laguerre polynomials

g colored partition algebras as centralizer algebras of wreath products abstract let g be a group we define an associative algebra pk x g that is a partition algebra whose diagrams have oriented edges labeled by elements of g for g finite we show that pk x g is the centralizer algebra of an action of the wreath product g s n on tensor powers of its permutation module

universality for orthogonal and symplectic laguerre type ensembles we give a proof of the universality conjecture for orthogonal 1 and symplectic 4 random matrix ensembles of laguerre type in the bulk of the spectrum as well as at the hard and soft spectral edges our results are stated precisely in the introduction theorems 1 1 1 4 1 6 and corollaries 1 2 1 5 1 7 they concern the appropriately rescaled kernels k n correlation and cluster functions gap probabilities and the distributions of the largest and smallest eigenvalues corresponding results for unitary 2 laguerre type ensembles have been proved by the fourth author in ref 23 the varying weight case at the hard spectral edge was analyzed in ref 13 for 2 in this paper we do not consider varying weights our proof follows closely the work of the first two authors who showed in refs 7 8 analogous results for hermite type ensembles as in refs 7 8 we use the version of the orthogonal polynomial method presented in refs 22 25 to analyze the local eigenvalue statistics the necessary asymptotic information on the laguerre type orthogonal polynomials is taken from ref 23

the problem of harmonic analysis on the infinite dimensional unitary group abstract the goal of harmonic analysis on a noncommutative group is to decompose the most natural unitary representations of this group like the regular representation on irreducible ones the infinite dimensional unitary group u is one of the basic examples of big groups whose irreducible representations depend on infinitely many parameters our aim is to explain what the harmonic analysis on u consists of we deal with unitary representations of a reasonable class which are in 1 1 correspondence with characters central positive definite normalized functions on u the decomposition of any representation of this class is described by a probability measure called spectral measure on the space of indecomposable characters the indecomposable characters were found by dan voiculescu in 1976 the main result of the present paper consists in explicitly constructing a 4 parameter family of natural representations and computing their characters we view these representations as a substitute of the nonexisting regular representation of u we state the problem of harmonic analysis on u as the problem of computing the spectral measures for these natural representations a solution to this problem is given in the next paper harmonic analysis on the infinite dimensional unitary group and determinantal point processes math 0109194 to appear in ann math joint with alexei borodin we also prove a few auxiliary general results in particular it is proved that the spectral measure of any character of u can be approximated by a sequence of discrete spectral measures for the restrictions of the character to the compact unitary groups u n this fact is a starting point for computing spectral measures

a tower construction for the radical in brauer s centralizer algebras abstract in this paper we study the structure of the brauer centralizer algebras in the case that the multiplication constant x is a rational integer several authors have studied the structure of these algebras for generic values of x in particular hans wenzl showed that the brauer algebras can be obtained from jones basic construction and he used that fact to prove that the brauer algebras are semisimple when x is not a rational integer the tower construction is a method to study towers of semisimple algebras hence it does not apply in our case where the algebras in the tower eventually have radicals our first step in this paper is to modify the tower construction so that it does a simultaneous tower construction of the radicals and the semisimple quotients of a tower of algebras the rest of the paper is devoted to describing these constructions explicitly in the brauer algebra case one surprising corollary of this method is a connection between two seemingly distinct criteria for the simplicity of certain subrings of the brauer algebras it is possible to identify explicitly certain subrings of the brauer algebras which are the matrix rings corresponding to irreducible representations in the semisimple case in previous work these authors gave a combinatorial condition for simplicity of these individual matrix rings when x is a rational integer the tower construction gives a second algebraic condition for simplicity it is difficult to see why these two conditions are equivalent however the methods used in this paper make that clear

symmetric and exterior powers linear source modules and representations of schur superalgebras we study modules for the general linear group over an infinite field of arbitrary characteristic which are direct summands of tensor products of exterior powers and symmetric powers of the natural module these modules which we call listing modules include the tilting modules and the injective modules for schur algebras the modules are studied via their relationship to linear source modules for symmetric groups on the one hand and simple modules for schur superalgebras on the other listing modules are parametrized by certain pairs of partitions they are used to describe by generators and relations the grothendieck ring of polynomial functors generated by the symmetric and exterior powers we also continuing work of j grabmeier describe the vertices and sources of linear source modules for symmetric groups 2000 mathematical subject classification 20g05 20c30

the quasi invariance property for the gamma kernel determinantal measure abstract the gamma kernel is a projection kernel of the form a x b y b x a y x y where a and b are certain functions on the one dimensional lattice expressed through euler s function the gamma kernel depends on two continuous parameters its principal minors serve as the correlation functions of a determinantal probability measure p defined on the space of infinite point configurations on the lattice as was shown earlier a borodin g olshanski adv math 194 2005 141 202 arxiv math ph 0305043 p describes the asymptotics of certain ensembles of random partitions in a limit regime theorem the determinantal measure p is quasi invariant with respect to finitary permutations of the nodes of the lattice this result is motivated by an application to a model of infinite particle stochastic dynamics

semiclassical asymptotics of orthogonal polynomials riemann hilbert problem and universality in the matrix model we derive semiclassical asymptotics for the orthogonal polynomials pn z on the line with respect to the exponential weight exp inv z where v z is a double well quartic polynomial in the limit when n n 1 we assume that n n cri for some 0 where cr is the critical value which separates orthogonal polynomials with two cuts from the ones with one cut simultaneously we derive semiclassical asymptotics for the recursive coe cients of the orthogonal polynomials and we show that these coe cients form a cycle of period two which drifts slowly with the change of the ratio n n the proof of the semiclassical asymptotics is based on the methods of the theory of integrable systems and on the analysis of the appropriate matrix riemannhilbert problem as an application of the semiclassical asymptotics of the orthogonal polynomials we prove the universality of the local distribution of eigenvalues in the matrix model with the double well quartic interaction in the presence of two cuts contents

random partitions and the gamma kernel abstract we study the asymptotics of certain measures on partitions the so called z measures and their relatives in two different regimes near the diagonal of the corresponding young diagram and in the intermediate zone between the diagonal and the edge of the young diagram we prove that in both cases the limit correlation functions have determinantal form with a correlation kernel which depends on two real parameters in the first case the correlation kernel is discrete and it has a simple expression in terms of the gamma functions in the second case the correlation kernel is continuous and translationally invariant and it can be written as a ratio of two suitably scaled hyperbolic sines

combinatorial and topological approach to the 3d ising model we extend the planar pfaffian formalism for the evaluation of the ising partition function to lattices of high topological genus g the 3d ising model on a cubic lattice where g is proportional to the number of sites is discussed in detail the expansion of the partition function is given in terms of 22g pfaffians classified by the oriented homology cycles of the lattice i e by its spin structures correct counting is guaranteed by a signature term which depends on the topological intersection of the oriented cycles through a simple bilinear formula the role of a gauge symmetry arising in the above expansion is discussed the same formalism can be applied to the counting problem of perfect matchings over general lattices and provides a determinant expansion of the permanent of 0 1 matrices

on partition algebras for complex reflection groups abstract in this paper we study the representation theory of two partition algebras related to complex reflection groups the colored partition algebras p k n r introduced by bloss m bloss g colored partition algebras as centralizer algebras of wreath products j algebra 265 2003 690 710 and the algebras t k n r introduced by tanabe k tanabe on the centralizer algebra of the unitary reflection group g m p n nagoya math j 148 1997 113 126 in particular we describe the decomposition of these algebras in terms of irreducible representations

characters of algebras containing a jones basic construction the temperley lieb okada brauer and birman wenzl algebras abstract we begin by determining in a general form the characters of irreducible representations of a jones basic construction and use this result to compute the characters of the temperley lieb algebras and the okada algebras in the case of the birman wenzl algebra some of the characters are determined by the general theorem and the others are computed by using the duality between the birman wenzl algebras and the drinfeld jimbo quantum groups of types b and c the computations involve certain characters of the quantum group these are polynomials invariant under the weyl group of type b we are able to decompose these weyl group symmetric functions and obtain a combinatorial rule for computing the irreducible characters of the birman wenzl algebras the combinatorial rules for computing the irreducible characters of the iwahori hecke algebras and the combinatorial rule for computing the irreducible characters of the brauer algebras are both special cases of our rule for the birman wenzl algebras

stochastic higher spin six vertex model and macdonald measures we prove an identity that relates the q laplace transform of the height function of a higher spin inhomogeneous stochastic six vertex model in a quadrant on one side and a multiplicative functional of a macdonald measure on the other the identity is used to prove the gue tracy widom asymptotics for two instances of the stochastic six vertex model via asymptotic analysis of the corresponding schur measures

integrable operators and the squares of hankel operators integrable operators arise in random matrix theory where they describe the asymptotic eigenvalue distribution of large self adjoint random matrices from the generalized unitary ensembles this paper gives sufficient conditions for an integrable operator to be the square of a hankel operator and applies the condition to the airy associated laguerre modified bessel and whittaker functions

asymptotics of plancherel measures for symmetric groups 1 1 plancherel measures given a finite group g by the corresponding plancherel measure we mean the probability measure on the set g of irreducible representations of g which assigns to a representation g the weight dim g for the symmetric group s n the set s n is the set of partitions of the number n which we shall identify with young diagrams with n squares throughout this paper the plancherel measure on partitions arises naturally in representation theoretic combinatorial and probabilistic problems for example the plancherel distribution of the first part of a partition coincides with the distribution of the longest increasing subsequence of a uniformly distributed random permutation 31 we denote the plancherel measure on partitions of n by mn