integration of quantum many body problems by affine knizhnik zamolodchikov equations abstract an equivalence of the affine analogues of knizhnik zamolodchikov and dunkl equations is established for arbitrary root systems as a corollary an isomorphism is obtained from the solutions of the above kz equations in the induced representations of hecke algebras to the eigenfunctions for the corresponding matrix quantum calogero sutherland operators which generalizes the matsuo map

yang s system of particles and hecke algebras the graded hecke algebra has a simple realization as a certain algebra of operators acting on a space of smooth functions this operator algebra arises from the study of the root system analogue of yang s system of n particles on the real line with delta function potential it turns out that the spectral problem for this generalization of yang s system is related to the problem of finding the spherical tempered representations of the graded hecke algebra this observation turns out to be very useful for both these problems application of our technique to affine hecke algebras yields a simple formula for the formal degree of the generic iwahori spherical discrete series representations

level dynamics in pseudointegrable billiards an experimental study abstract the level dynamics of pseudointegrable systems with different genus numbers g is studied experimentally using microwave cavities for higher energies the distribution of the eigenvalue velocities is gaussian as it is expected for chaotic systems with time reversal symmetry and shows no dependence on g also the curvature distribution p k for large k is decaying as it is expected for chaotic systems i e p k k 3 for small k an intermediate behavior is found where p k changes from integrable towards chaotic behavior with growing g

affine yangians and deformed double current algebras in type a we study the structure of yangians of affine type and deformed double current algebras which are deformations of the enveloping algebras of matrix w1 8 algebras we prove that they admit a pbw type basis establish a connection limit construction between these two types of algebras and toroidal quantum algebras and we give three equivalent definitions of deformed double current algebras we construct a schur weyl functor between these algebras and rational cherednik algebras

some conjectures and results concerning the homology of nilpotent lie algebras in this paper we consider the lie algebra homology of for l a complex lie algebra our goal is to express the homology of l in terms of the homology of l this problem comes up in previous work by the author on the macdonald root system conjectures we present a number of conjectures related to this problem the simplest of these conjectures asserts that h l h l l i

a unification of knizhnik zamolodchikov and dunkl operators via affine hecke algebras some generalizations of the lusztig lascoux schutzenberger operators for affine hecke algebras are considered as corollaries we obtain lusztig s isomorphisms from affine hecke algebras to their degenerate versions a natural interpretation of the dunkl operators and a new class of differential difference operators generalizing dunkl s ones and the knizhnik zamolodchikov operators from the two dimensional conformal field theory

toroidal and level 0 uq sln 1 actions on uq gln 1 modules 1 utilizing a braid group action on a completion of uq sln 1 an algebra homomorphism from the toroidal algebra uq sln 1 tor n 2 to a completion of uq gln 1 is obtained 2 the toroidal actions by saito induces a level 0 uq sln 1 action on level 1 integrable highest weight modules of uq sln 1 another level 0 uq sln 1 action was defined by jimbo et al in the case n 1 using the fact that the intertwiners of uq sln 1 modules are intertwiners of toroidal modules for an appropriate comultiplication the relation between these two level 0 uq sln 1 actions is clarified

cyclic homology and the macdonald conjectures leta k denote the ring t t k 1 and letg be a reductive complex lie algebra with exponentsm 1 m n this paper concerns the lie algebra cohomology ofg a k considered as a bigraded algebra here one of the gradings is homological degree and the other which we callweight is inherited from the obvious grading ofg a k we conjecture that this lie algebra cohomology is an exterior algebra withk 1 generators of homological degree 2m s 1 fors 1 2 n of thesek 1 generators of degree 2m s 1 one has weight 0 and the others have weights k 1 m s t fort 1 2 k it is shown that this conjecture about the lie algebra cohomology of a k implies the macdonald root system conjectures next we consider the case thatg is a classical lie algebra with root systema n b n c n ord n it is shown that our conjecture holds in the limit onn asn approaches infinity which amounts to the computation of the cyclic and dihedral cohomologies ofa k lastly we discuss the relevance of this limiting case to the case of finiten in this situation

level statistics and eigenfunctions of square torus billiards manifesting the transition from regular to chaotic behaviors we thoroughly analyze the level statistics and eigenfunctions in concentric as well as nonconcentric square torus billiards we confirm the characteristics of quantum and classical correspondence and the existence of scarred and superscarred modes in concentric square torus billiards furthermore we not only verify that the transition from regular to chaotic behaviors can be manifested in nonconcentric square torus billiards but also develop an analytical distribution to excellently fit the numerical level statistics finally we intriguingly observe that numerous eigenstates commonly exhibit the wave patterns to be an ensemble of classical diamond trajectories as the effective wavelengths are considerably shorter than the size of internal hole

analytically solvable dynamical systems which are not integrable abstract the central two fold aim of this paper is to expose a class of nonintegrable systems whose solutions can be analytically expressed in terms of computationally tractable algorithms and to explicitly construct a demonstrably meaningful general solution for one member of this class toward this end information theoretic results are reviewed which show that dynamical systems of null metric entropy have general solutions of null algorithmic complexity hence all systems exhibiting null metric entropy including ergodic only and mixing only systems have orbits which can be specified by algorithmically meaningful analytic expressions whose information content is very much less than that of the orbits they describe in consequence we may quite legitimately define the class of systems having null metric entropy ot be algorithmically integrable called a integrable herein after establishing the existence of a integrability this paper turns to the problem of exposing a specific nonintegrable but nonetheless a integrable system whose algorithmically meaningful solution can be explicitly derived the billiard moving within a plane polygon having the shape of a 60 120 rhombus is shown to be a simple example indeed the whole class of billiards moving in polygons whose angles are all rational multiples of have null metric entropy and are therefore a integrable systems moreover all rational billiards appear to have explicitly derivable solutions as is discussed herein and in a recent preprint by bjorn birnir because rational billiards are dense in the set of all plane billiards including chaotic ones the present paper suggests that a integrable billiards may provide a long sought analytic route to chaos finally it is pointed out that null entropied a integrable systems are perhaps the broadest category of systems which can reasonably be termed integrable for systems having positive metric entropy are truly nonintegrable due to the presence of orbits which are their own simplest description and their own fastest computer

t equivariant k theory of generalized flag varieties let g be a kac moody group with borel subgroup b and compact maximal torus t analogous to kostant and kumar kostant b kumar s 1986 proc nati acad sci usa 83 1543 1545 we define a certain ring y purely in terms of the weyl group w associated to g and its action on t by dualizing y we get another ring 1 which we prove is canonically isomorphic with the t equivariant ktheory kt g b of gib now kt g b apart from being an algebra over kt pt a t also has a weyl group action and moreover kt g b admits certain operators dw w similar to the demazure operators defined on a t we prove that these structures on kt g b come naturally from the ring y by evaluating the a t module w at 1 we recover k g b together with the above mentioned structures we believe that many of the results of this paper are new in the finite case i e g is a finite dimensional semisimple group over c

periodic orbit theory in fractal drums abstract the level statistics of pseudointegrable fractal drums is studied numerically using periodic orbit theory we find that the spectral rigidity 3 l which is a measure for the correlations between the eigenvalues decreases to quite small values as compared to systems with only small boundary roughness thereby approaching the behavior of chaotic systems the periodic orbit results are in good agreement with direct calculations of 3 l from the eigenvalues

a new generalisation of macdonald polynomials we introduce a new family of symmetric multivariate polynomials whose coefficients are meromorphic functions of two parameters q t and polynomial in a further two parameters u v we evaluate these polynomials explicitly as a matrix product at u v 0 they reduce to macdonald polynomials while at q 0 u v s they recover a family of inhomogeneous symmetric functions originally introduced by borodin

range of spectral correlations in pseudointegrable systems gaussian orthogonal ensemble statistics in a rectangular membrane with a point scatterer conventional wisdom holds that a finite reverberant system with chaotic ray trajectories will have at high frequencies eigenvalue statistics identical to those of the gaussian orthogonal ensemble goe of random matrices it also holds that a nonchaotic system will have simple poissonian statistics recent experiments on the eigenvalues of elastic blocks with angled cuts and recent calculations of the eigenfrequencies of membranes with staircaselike jagged boundaries and the eigenfrequencies of a rectangular domain with a single isotropic point scatterer have however found goe statistics even in these pseudointegrable systems even though all rays in such systems are nonchaotic in this work the rectangular domain with a single isotropic point scatterer is studied further in contrast to recent related work the scatterer is characterized here by its t matrix and scattering cross section it is shown that the long range level repulsion in this system is not in precise accord with the predictions of the goe nor is the long range spectral rigidity goe does though correctly describe the short range statistics a quantitative prediction for the range in which goe applies is advanced based upon the lifetime of a ray against mixing i e based upon the scattering cross section of the scatterer this prediction is corroborated by numerical calculations of the eigenfrequencies

yangian gelfand zetlin bases gl n jack polynomials and computation of dynamical correlation functions in the spin calogero sutherland model we consider the invariant calogero sutherland models with n 1 2 3 in the framework of symmetric polynomials

periodic orbit theory and spectral rigidity in pseudointegrable systems we calculate numerically the periodic orbits of pseudointegrable systems of low genus numbers g that arise from rectangular systems with one or two salient corners from the periodic orbits we calculate the spectral rigidity delta3 l using semiclassical quantum mechanics with l reaching up to quite large values we find that the diagonal approximation is applicable when averaging over a suitable energy interval comparing systems of various shapes we find that our results agree well with delta3 calculated directly from the eigenvalues by spectral statistics therefore additional terms such as e g diffraction terms seem to be small in the case of the systems investigated in this work by reducing the size of the corners the spectral statistics of our pseudointegrable systems approaches that of an integrable system whereas very large differences between integrable and pseudointegrable systems occur when the salient corners are large both types of behavior can be well understood by the properties of the periodic orbits in the system

quantum knizhnik zamolodchikov equations and affine root systems quantum difference knizhnik zamolodchikov equations s1 fr are generalized for ther matrices from ch1 with the arguments in arbitrary root systems and their formal counterparts in particular qkz equations with certain boundary conditions are introducted the self consistency of the equations from fr and the cross derivative integrability conditions for ther matrix kz equations from ch2 are obtained as corollaries a difference counterpart of the quantum many body problem connected with macdonald s operators is defined as an application

energy fluctuations of pseudointegrable systems with growing surface roughness the eigenfrequencies of two dimensional systems with fractal boundaries and with nonscaling rough boundaries are calculated numerically by the lanczos algorithm and analyzed by means of level statistics the systems are pseudointegrable and the fluctuations of their eigenvalue spectra show a global statistical behavior between the poisson and the wigner distributions with increasing irregularity of the boundary the systems approach the wigner limit and the results seem to depend only on the genus number of the geometry and not on details such as the asymptotic shape of the geometry the type of roughness scaling or nonscaling and the boundary conditions neumann or dirichlet no transition between localized and extended states is found in fractal drums

geometric representations of graded and rational cherednik algebras abstract we provide geometric constructions of modules over the graded cherednik algebra h gr and the rational cherednik algebra h rat attached to a simple algebraic group g together with a pinned automorphism these modules are realized on the cohomology of affine springer fibers of finite type that admit c actions in the rational cherednik algebra case the standard grading on these modules is derived from the perverse filtration on the cohomology of affine springer fibers coming from its global analog hitchin fibers when is trivial we show that our construction gives the irreducible finite dimensional spherical modules l triv of h gr and of h rat we give a formula for the dimension of l triv and give a geometric interpretation of its frobenius algebra structure the rank two cases are studied in further details

level statistics and eigenfunctions of pseudointegrable systems dependence on energy and genus number we study the level statistics second half moment i 0 and rigidity ensuremath delta 3 and the eigenfunctions of pseudointegrable systems with rough boundaries of different genus numbers g we find that the levels form energy intervals with a characteristic behavior of the level statistics and the eigenfunctions in each interval at low enough energies the boundary roughness is not resolved and accordingly the eigenfunctions are quite regular functions and the level statistics shows poisson like behavior at higher energies the level statistics of most systems moves from poisson like toward wigner like behavior with increasing g on investigating the wave functions we find many chaotic functions that can be described as a random superposition of regular wave functions the amplitude distribution p ensuremath psi of these chaotic functions was found to be gaussian with the typical value of the localization volume v mathrm loc ensuremath approx 0 33 for systems with periodic boundaries we find several additional energy regimes where i 0 is relatively close to the poisson limit in these regimes the eigenfunctions are either regular or localized functions where p ensuremath psi is close to the distribution of a sine or cosine function in the first case and strongly peaked in the second case an interesting intermediate case between chaotic and localized eigenfunctions also appears

influence of boundary conditions on level statistics and eigenstates at the metal insulator transition abstract we investigate the influence of the boundary conditions on the scale invariant critical level statistics at the metal insulator transition of disordered three dimensional orthogonal and two dimensional unitary and symplectic tight binding models the distribution of the spacings between consecutive eigenvalues is calculated numerically and shown to be different for periodic and dirichlet boundary conditions whereas the critical disorder remains unchanged the peculiar correlations of the corresponding critical eigenstates leading to anomalous diffusion seem not to be affected by the change of the boundary conditions

macdonald s constant term conjectures for exceptional root systems we announce proofs of macdonald s constant term conjectures for the affine root systems s f 4 and s f 4 v we also give an algorithm for deciding the conjectures for the remaining root systems s e 6 s e 7 and s e 8 and prove that the constant term in question can indeed be expressed in closed form combined with previous work of zeilberger bressoud kadell and gustafson our results imply that macdonald s conjectures are true in form for any root system

statistical measures for eigenfunctions of nonseparable quantum billiard systems abstract we study two statistical measures for eigenfunctions of several classically pseudointegrable and chaotic billiard systems the directional energy distribution and the nodal line curvature distribution a sharp distinction between the classically pseudointegrable and the chaotic systems is possible with both of these measures they can also be used to distinguish between regular and irregular eigenfunctions of the billiard systems using the same data set additionally the amplitude distribution is calculated for single regular and irregular states and accumulated over several hundred states the well known gaussian prediction holds well for irregular wave functions of the pseudointegrable system too

periodic orbits and spectral statistics of pseudointegrable billiards we demonstrate for a generic pseudointegrable billiard that the number of periodic orbit families with length less than l increases as frac ensuremath pi b 0 l 2 a l where b 0 is a constant and a l is the average area occupied by these families we also find that a l increases with l before saturating finally we show that periodic orbits provide a good estimate of spectral correlations in the corresponding quantum spectrum and thus conclude that diffraction effects are not as significant in such studies

group actions on algebras and the graded lie structure of hochschild cohomology hochschild cohomology governs deformations of algebras and its graded lie structure plays a vital role we study this structure for the hochschild cohomology of the skew group algebra formed by a nite group acting on an al gebra by automorphisms we examine the gerstenhaber bracket with a view toward deformations and developing bracket formulas we then focus on the linear group actions and polynomial algebras that arise in orbifold theory and representation theory deformations in this context include graded hecke alge bras and symplectic reection algebras we give some general results describing when brackets are zero for polynomial skew group algebras which allow us in particular to nd noncommutative poisson structures for abelian groups we express the bracket using inner products of group characters lastly we inter pret results for graded hecke algebras

the homflypt skein algebra of the torus and the elliptic hall algebra we give a generators and relations presentation of the homflypt skein algebra h of the torus t2 and we give an explicit description of the module corresponding to the solid torus using this presentation we show that h is isomorphic to the 1 specialization of the elliptic hall algebra of burban and schiffmann as an application for an iterated cable k of the unknot we use the elliptic hall algebra to construct a 3 variable polynomial that specializes to the colored homflypt polynomial of k we show that this polynomial also specializes to one constructed by cherednik and danilenko using the gln double affine hecke algebra this proves one of the connection conjectures in their recent work

a low dimensional semidefinite relaxation for the quadratic assignment problem the quadratic assignment problem qap is arguably one of the hardest np hard discrete optimization problems problems of dimension greater than 25 are still considered to be large scale current successful solution techniques use branch and bound methods which rely on obtaining strong and inexpensive bounds in this paper we introduce a new semidefinite programming sdp relaxation for generating bounds for the qap in the trace formulation we apply majorization to obtain a relaxation of the orthogonal similarity set of the quadratic part of the objective function this exploits the matrix structure of qap and results in a relaxation with much smaller dimension than other current sdp relaxations we compare the resulting bounds with several other computationally inexpensive bounds such as the convex quadratic programming relaxation qpb we find that our method provides stronger bounds on average and is adaptable for branch and bound methods

orthogonality relations and cherednik identities for multivariable baker akhiezer functions abstract we establish orthogonality relations for the baker akhiezer ba eigenfunctions of the macdonald difference operators we also obtain a version of cherednik macdonald mehta integral for these functions as a corollary we give a simple derivation of the norm identity and cherednik macdonald mehta integral for macdonald polynomials in the appendix written by the first author we prove a summation formula for ba functions we also consider more general identities of cherednik type which we use to introduce and construct more general twisted ba functions this leads to a construction of new quantum integrable models of macdonald ruijsenaars type

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

discrete series characters for affine hecke algebras and their formal degrees we introduce the generic central character of an irreducible discrete series representation of an affine hecke algebra using this invariant we give a new classification of the irreducible discrete series characters for all abstract affine hecke algebras except for the types e n 1 n 6 7 8 with arbitrary positive parameters and we prove an explicit product formula for their formal degrees in all cases

matrix product formula for macdonald polynomials we derive a matrix product formula for symmetric macdonald polynomials our results are obtained by constructing polynomial solutions of deformed knizhnik zamolodchikov equations which arise by considering representations of the zamolodchikov faddeev and yang baxter algebras in terms of t deformed bosonic operators these solutions are generalised probabilities for particle configurations of the multi species asymmetric exclusion process and form a basis of the ring of polynomials in n variables whose elements are indexed by compositions for weakly increasing compositions anti dominant weights these basis elements coincide with non symmetric macdonald polynomials our formulas imply a natural combinatorial interpretation in terms of solvable lattice models they also imply that normalisations of stationary states of multi species exclusion processes are obtained as macdonald polynomials at q 1

irreducible representations of the rational cherednik algebra associated to the coxeter group h3 abstract this paper describes irreducible representations in category o of the rational cherednik algebra h c h 3 h associated to the exceptional coxeter group h 3 and any complex parameter c we compute the characters of all these representations explicitly as a consequence we classify all the finite dimensional irreducible representations of h c h 3 h

twisted strong macdonald theorems and adjoint orbits the strong macdonald theorems state that for l reductive and s an odd variable the cohomology algebras h l z z n and h l z s are freely generated and describe the cohomological s and z degrees of the generators the resulting identity for the z weighted euler characteristic is equivalent to macdonald s constant term identity for a finite root system we calculate h mathfrak p z n mathfrak p and h mathfrak p s for mathfrak p a standard parahoric in a twisted loop algebra giving strong macdonald theorems that take into account both a parabolic component and a possible diagram automorphism twist in particular we show that h mathfrak p z n mathfrak p contains a parabolic subalgebra of the coinvariant algebra of the fixed point subgroup of the weyl group of l and thus is no longer free we also prove a strong macdonald theorem for h mathfrak b s mathfrak n and h mathfrak b z n mathfrak n when mathfrak b and mathfrak n are iwahori and nilpotent subalgebras respectively of a twisted loop algebra for each strong macdonald theorem proved taking z weighted euler characteristics gives an identity equivalent to macdonald s constant term identity for the corresponding affine root system as part of the proof we study the regular adjoint orbits for the adjoint action of the twisted arc group associated to l proving an analogue of the kostant slice theorem

double affine hecke algebras and macdonald s conjectures in this paper we prove the main results about the structure of double affine hecke algebras announced in c1 c2 the technique is based on the realization of these algebras in terms of demazure lusztig operators bgg d l2 ls c3 and rather standard facts from the theory of affine weyl groups in particular it completes the proof partially published in c2 of the macdonald scalar product conjecture see ml 12 6 including the famous macdonald constant term conjecture the q t case we mainly follow the opdam paper 0 where the macdonald mehta conjectures in the degenerate differential case were deduced from certain properties of the heckman opdam operators ho and the existence of the shift operators heckman s interpretation of these operators via the so called dunkl operators see he and also c5 was important to our approach we note that the ho operators are closely related to the so called quantum many body problem calogero sutherland moser olshanetsky perelomov the conformal field theory knizhnik zamolodchikov equations the harmonic analysis on symmetric spaces harish chandra helgason etc and last but not the least the classic theory of the hypergeometric functions establishing the connection between the difference counterparts of heckman opdam operators introduced in c4 and the macdonald theory ml m2 including the construction of the difference shift operators is the main thrust of this paper once the connection is established it is not very difficult to calculate the scalar squares of the macdonald polynomials and to prove the constant term conjecture from his fundamental paper m3 to simplify the exposition we discuss the reduced root systems only and impose the relation q tk for k e z to avoid infinite products in the definition of macdonald s pairing the purpose of this work is to present a concrete application of the new technique arbitrary q t can be handled in a

bosonic and fermionic eigenstates for generalized sutherland models we construct bosonic and fermionic eigenstates for the generalized sutherland models associated with arbitrary reduced root systems respectively through w symmetrization and w anti symmetrization of heckman opdam s nonsymmetric jacobi polynomials square norms of the nonsymmetric heckman opdam polynomials are evaluated from their rodrigues formulae the w symmetrization and w anti symmetrization of the nonsymmetric polynomials enable us to evaluate square norms of bosonic and fermionic eigenstates for the generalized sutherland models

pseudointegrable systems in classical and quantum mechanics abstract we study particles moving in planar polygonal enclosures with rational angles and show by several methods that trajectories in the classical phase space explore two dimensional invariant surfaces which are generically not tori as in integrable systems but instead have the topology of multiply handled spheres the quantum mechanics of one such pseudointegrable system is studied in detail by computing energy levels using an exact formalism this system consists of motion on a unit coordinate torus containing a square reflecting obstacle with side l we find that neighbouring levels avoid degeneracies as l varies and that the probability distribution for the spacing s of adjacent levels vanishes linearly as s 0 level repulsion the weyl area rule plus edge and corner corrections gives a very accurate approximation for the mean level density oscillatory corrections to the mean level density are given as a sum over closed classical paths for pseudointegrable systems these closed paths form families covering part of the phase space invariant surfaces