second quantization of the elliptic calogero sutherland model we construct a quantum field theory model of anyons on a circle and at finite temperature we find an anyon hamiltonian providing a second quantization of the elliptic calogero sutherland model this allows us to prove a remarkable identity which is a starting point for an algorithm to construct eigenfunctions and eigenvalues of the elliptic calogero sutherland hamiltonian

source identity and kernel functions for inozemtsev type systems the inozemtsev hamiltonian is an elliptic generalization of the differential operator defining the bcn trigonometric quantum calogero sutherland model and its eigenvalue equation is a natural many variable generalization of the heun differential equation we present kernel functions for inozemtsev hamiltonians and chalykh feigin veselov sergeev type deformations thereof our main result is a solution of a heat type equation for a generalized inozemtsev hamiltonian which is the source of all these kernel functions applications are given including a derivation of simple exact eigenfunctions and eigenvalues of the inozemtsev hamiltonian

some combinatorial properties of jack symmetric functions if 1 l n then write 2 n or 1 n if p is another partition then write p c j if pi 6 for all i i e if the diagram of 1 contains the diagram of p if ipl ii 1 then write p 2 1 reverse lexicographic order if either j i or the first nonvanishing difference e pi is positive for instance writing 2 1 i for ai i it 5 5 41 32 5 311 221 2111 11111 finally write p i if ipl lal and p1 pl2 pli a1 a2 li for all i macdonald m p 63 call the partial ordering the natural ordering but we will call it the dominance ordering let x x x2 be an infinite set of indeterminates as in mi we

explicit formulae for the generalized hermite polynomials in superspace we provide explicit formulae for the orthogonal eigenfunctions of the supersymmetric extension of the rational calogero moser sutherland model with harmonic confinement i e the generalized hermite or hi jack polynomials in superspace the construction relies on the triangular action of the hamiltonian on the supermonomial basis this translates into determinantal expressions for the hamiltonian s eigenfunctions

boundary conditions for integrable quantum systems a new class of boundary conditions is described for quantum systems integrable by means of the quantum inverse scattering r matrix method the method proposed allows the author to treat open quantum chains with appropriate boundary terms in the hamiltonian the general considerations are applied to the xxz and xyz models the nonlinear schrodinger equation and toda chain

dold kan type theorems and morita equivalences of functor categories abstract the main results of the paper describe conditions which imply morita equivalences of certain functor categories it is shown that various well known results of homological algebra which are related to morita equivalences can be obtained as a specialization of these results a new application is obtained in the case of two categories associated to an operad

testing hall post inequalities with exactly solvable n body problems the hall post inequalities provide lower bounds on n body energies in terms of n body energies with n n they are rewritten and generalized to be tested with exactly solvable models of calogero sutherland type in one and higher dimensions the bound for n spinless fermions in one dimension is better saturated at large coupling than for noninteracting fermions in an oscillator potential

some multidimensional integrals related to many body systems with the 1 r2 potential an n dimensional integral evaluated by k aomoto 1988 is shown to represent the density matrix for an impurity particle in the 1 r2 quantum many body problem on a line the value of the n dimensional integral representing the same density matrix in periodic boundary conditions is conjectured as too is the value of an n dimensional integral which represents a two point correlation function in the system also the partition function of a related classical hamiltonian is evaluated by formulating a conjecture which asserts that the sum of jacobians of a certain change of variables in n dimensions is a constant

complete integrability of relativistic calogero moser systems and elliptic function identities poincare invariant generalizations of the galilei invariant calogero moser a particle systems are studied a quantization of the classical integrals s1 sn is presented such that the operators si9 sn mutually commute as a corollary it follows that si9 9sn poisson commute these results hinge on functional equations satisfied by the weierstrass and 0 functions a generalized cauchy identity involving the function leads to an n x n matrix l whose symmetric functions are proportional to sl5 9sn

correlation functions of the bc calogero sutherland model the bc type calogero sutherland model csm is an integrable extension of the ordinary a type csm that possesses a reflection symmetry point the bc cs mi sr elated to the chiral classes of random matrix ensembles rmes in exactly the same way as the a csm is related to the dyson classes we first develo pt he fermionic replica model formalism suitable to treat all chiral rmes by exploiting generalized colour flavour transformation we then extend the method to find the exact asymptotics of the bc csm density profile consistency of our result with the c 1g aussian conformal field theory description is verified the emerging friedel oscillations structure and sum rules are discussed in details we also compute the distribution of the particle nearest to the reflection point

the calogero sutherland model and generalized classical polynomials multivariable generalizations of the classical hermite laguerre and jacobi polynomials occur as the polynomial part of the eigenfunctions of certain schrodinger operators for calogero sutherland type quantum systems for the generalized hermite and laguerre polynomials the multidimensional analogues of many classical results regarding generating functions differentiation and integration formulas recurrence relations and summation theorems are obtained we use this and related theory to evaluate the global limit of the ground state density obtaining in the hermite case the wigner semi circle law and to give an explicit solution for an initial value problem in the hermite and laguerre case

twisted relative trace formulae with a view towards unitary groups we introduce a twisted relative trace formula which simultaneously generalizes the twisted trace formula of langlands et al in the quadratic case and the relative trace formula of jacquet and lai certain matching statements relating this twisted relative trace formula to a relative trace formula are also proven including the relevant fundamental lemma in the biquadratic case using recent work of jacquet lapid and their collaborators and the rankin selberg integral representation of the asai l function obtained by flicker using the theory of jacquet piatetskii shapiro and shalika we give the following application let e f be a totally real quadratic extension with gal e f let u be a quasi split unitary group with respect to a cm extension m f and let u res e f u under suitable local hypotheses we show that a cuspidal cohomological automorphic representation of u whose asai l function has a pole at the edge of the critical strip is nearly equivalent to a cuspidal cohomological automorphic representation of u that is u distinguished in the sense that there is a form in the space of admitting a nonzero period over u this provides cohomologically nontrivial cycles of middle dimension on unitary shimura varieties analogous to those on hilbert modular surfaces studied by harder langlands and rapoport

some properties of the calogero sutherland model with reflections we prove that the calogero sutherland model with reflections the model possesses a property of duality relating the eigenfunctions of two hamiltonians with different coupling constants we obtain a generating function for their polynomial eigenfunctions the generalized jacobi polynomials the symmetry of the wavefunctions for certain particular cases associated with the root systems of the classical lie groups and is also discussed

a free energy bound for the hopfield model the author gives a simple upper and lower bound on the free energy density of the hopfield model of size n with p stored patterns in the limit where n and p tend to infinity with p n to alpha 1 the two bounds coincide for alpha 0

hardy littlewood varieties and semisimple groups we are interested in counting integer and rational points in affine algebraic varieties also under congruence conditions we introduce the notions of a strongly hardy littlewood variety and a relatively hardy littlewood variety in terms of counting rational points satisfying congruence conditions the definition of a strongly hardy littlewood variety is given in such a way that varieties for which the hardy littlewood circle method is applicable are strongly hardy littlewood

generic representation theory of finite fields in nondescribing characteristic abstract let rep f k denote the category of functors from finite dimensional f vector spaces to k modules where f is a field and k is a commutative ring we prove that if f is a finite field and char f is invertible in k then the k linear abelian category rep f k is equivalent to the product over all n 0 of the categories of k gl n f modules as a consequence if k is also a field then small projectives are also injective in rep f k and will have finite length even more is true if char k 0 the category rep f k will be semisimple in the last section we briefly discuss q 1 analogues and consider representations of various categories of finite sets the main result follows from a 1992 result by l g kovacs about the semigroup ring k m n f

yang baxter equation in long range interacting systems we consider the su p spin chains with long range interactions and the spin generalization of the calogero sutherland models we show that their properties derive from a transfer matrix obeying the yang baxter equation we obtain the expression of the conserved quantities of the dynamical models and we diagonalise them in the spin chain case we establish the connection between the degeneracies of the spectrum and the representation theory of the yangians we use a correspondence with the dynamical models to diagonalise the hamiltonian finally we extend the previous results to the case of a trigonometric r matrix

explicit formulae for the eigenfunctions of the n body calogero model we consider the quantum calogero model which describes n non distinguishable quantum particles on the real line confined by a harmonic oscillator potential and interacting via two body interactions proportional to the inverse square of the inter particle distance we elaborate a novel solution algorithm which allows us to obtain fully explicit formulae for its eigenfunctions arbitrary coupling parameter and particle number we also show that our method applies with minor changes to all calogero models associated with classical root systems

common algebraic structure for the calogero sutherland models we investigate a common algebraic structure for the rational and trigonometric calogero sutherland models by using the exchange operator formalism we show that the set of jack polynomials whose arguments are dunkl type operators provides an orthogonal basis for the rational case

hidden algebras of the super calogero and sutherland models we propose to parametrize the configuration space of one dimensional quantum systems of n identical particles by the elementary symmetric polynomials of bosonic and fermionic coordinates it is shown that in this parametrization the hamiltonians of the an bcn bn cn and dn calogero and sutherland models as well as their supersymmetric generalizations can be expressed for arbitrary values of the coupling constants as quadratic polynomials in the generators of a borel subalgebra of the lie algebra gl n 1 or the lie superalgebra gl n 1 n for the supersymmetric case these algebras are realized by first order differential operators this fact establishes the exact solvability of the models according to the general definition given by turbiner and implies that the calogero and jack sutherland polynomials as well as their supersymmetric generalizations are related to finite dimensional irreducible representations of the lie algebra gl n 1 and the lie superalgebra gl n 1 n

algebraic exact solvability of trigonometric type hamiltonians associated to root systems in this article we study and settle several structural questions concerning the exact solvability of the olshanetsky perelomov quantum hamiltonians corresponding to an arbitrary root system we show that these operators can be written as linear combinations of certain basic operators admitting infinite flags of invariant subspaces namely the laplacian and the logarithmic gradient of invariant factors of the weyl denominator the coefficients of the constituent linear combination become the coupling constants of the final model we also demonstrate the l2 completeness of the eigenfunctions obtained by this procedure and describe a straightforward recursive procedure based on the freudenthal multiplicity formula for constructing the eigenfunctions explicitly

a remarkable property of definite integrals we show that the integral of f u is the same as the integral of f x both with respect to x for a variety of functions u in this note f will denote any integrable function and all integral signs denote the cauchy principal value the formula oo 00 f u dx f x dx 00 oo where 2 u x 1 x goes back at least as far as cauchy and can be proved as follows from a graph of the relation 2 one sees that the solution for x has two branches 1 x y u iju2 4 oo t oo thus fo or00 r f u dx f u dx f u xi x du oo j0 oo but x x i 1 1 1 ju2 4 iu2 4 j thus for example exp x2 1 x2 dx exp x l x dx o 2e oo 1 r00 r2 w e x dx 2e2 oo 2e2 equations 1 2 clearly represent a useful property of the definite integral that deserves to be widely known what is more remarkable is that 1 remains true when 2 is replaced by v aj 3 u x2 y r j x s received june 21 1982 1980 mathematics subject classification primary 26a06 26a42 1983 american mathematical society 0025 5718 82 0000 0898 01 50 561 license or copyright restrictions may apply to redistribution see http www ams org journal terms of use

algorithms to solve the quantum sutherland model we give a self contained presentation and comparison of two different algorithms to explicitly solve quantum many body models of indistinguishable particles moving on a circle and interacting with two body potentials of 1 sin2 type the first algorithm is due to sutherland and well known the second one is a limiting case of a novel algorithm to solve the elliptic generalization of the sutherland model these two algorithms are different in several details we show that they are equivalent i e they yield the same solution and are equally simple

fi modules and stability for representations of symmetric groups in this paper we introduce and develop the theory of fi modules we apply this theory to obtain new theorems about the cohomology of the configuration space of n distinct ordered points on an arbitrary connected oriented manifold the diagonal coinvariant algebra on r sets of n variables the cohomology and tautological ring of the moduli space of n pointed curves the space of polynomials on rank varieties of n n matrices the subalgebra of the cohomology of the genus n torelli group generated by h1 and more the symmetric group sn acts on each of these vector spaces in most cases almost nothing is known about the characters of these representations or even their dimensions we prove that in each fixed degree the character is given for n large enough by a polynomial in the cycle counting functions that is independent of n in particular the dimension is eventually a polynomial in n in this framework representation stability in the sense of church farb for a sequence of sn representations is converted to a finite generation property for a single fi module