on the relaxation properties of stochastic ginzburg landau type models abstract considering the project of studying the basic properties of commonly used stochastic nonlinear dynamical systems we investigate and present new properties of the relaxation to equilibrium of stochastic ginzburg landau type models with polynomial interaction with nonlocal nonlinear terms in our approach we map the initial stochastic model on a imaginary time quantum field theory we show that with a suitable nonlocality a two particle bound state appears below the two particle threshold even if all the coefficients in the polynomial interaction are positive the existence of such a bound state changes the relaxation rate to equilibrium and so it is an experimentally observable effect of physical interest moreover we show that the bound state masses are sensitive to changes in the rate of nonlocality which may favor other phenomena related to the low lying spectrum of the dynamics generator our results involve a perturbative analysis supported by previous rigorous results in the computation of the bound state mass we use a bethe salpeter equation in the ladder approximation

component groups of unipotent centralizers in good characteristic abstract let g be a connected reductive group over an algebraically closed field of good characteristic for u g unipotent we describe the conjugacy classes in the component group a u of the centralizer of u our results extend work of the second author done for simple adjoint g over the complex numbers when g is simple and adjoint the previous work of the second author makes our description combinatorial and explicit moreover it turns out that knowledge of the conjugacy classes suffices to determine the group structure of a u thus we obtain the result previously known through case checking that the structure of the component group a u is independent of good characteristic

lax pair and darboux transformation of a noncommutative u n principal chiral model we present a noncommutative generalization of the lax formalism of the u n principal chiral model in terms of a one parameter family of flat connections the lax formalism is further used to derive a set of parametric noncommutative backlund transformations and an infinite set of conserved quantities from the lax pair we derive a noncommutative version of the darboux transformation of the model

character sheaves v this paper is part of a series s 131 devoted to the study of a class g of irreducible perverse sheaves called character sheaves on a connected reductive algebraic group g the numbering of chapters sections and references will continue that of s 131 this paper is a step towards the classification of character sheaves on g one of the main results is the following one under certain assumptions there is a natural surjective map with finite fibers from g to the set of all pairs 9 c up to conjugacy by the weyl group where 9 is a tame local system on the maximal torus and c is a two sided cell in the stabilizer w of 9 in the weyl group the assumptions made on g are

spin and wedge representations of infinite dimensional lie algebras and groups abstract we suggest a purely algebraic construction of the spin representation of an infinite dimensional orthogonal lie algebra sections 1 and 2 and a corresponding group section 4 from this we deduce a construction of all level one highest weight representations of orthogonal affine lie algebras in terms of creation and annihilation operators on an infinite dimensional grassmann algebra section 3 we also give a similar construction of the level one representations of the general linear affine lie algebra in an infinite dimensional wedge space along these lines we construct the corresponding representations of the universal central extension of the group sln k t t 1 in spaces of sections of line bundles over infinite dimensional homogeneous spaces section 5

approach to equilibrium of particles diffusing on curved surfaces abstract we present a simple numerical analysis of the diffusion on a curved surface given by the equation r 0 in a finite domain d r 3 the first non vanishing eigenvalue of the beltrami laplace operator with the reflecting boundary conditions is determined in our simulations for the p d g s s1 and i wp nodal periodic surfaces where d is their respective cubic unit cell we observe that the first eigenvalue for the surfaces of simple topology p d g i wp is smaller than for the surfaces of complex topology s s1

the spectrum of a coxeter transformation affine coxeter transformations and the defect map abstract the spectrum of the coxeter transformation r in a reflection group defined by a matrix a without odd cycles is determined in terms of spec a for affine a r determines the defect map which is given a simple formulation in terms of null roots the affine exponents determined by spec r are related to the coefficients of the fundamental co weight corresponding to the fork node of the coxeter dynkin diagram

one particle subspaces in the stochastic xy model we study the spectrum of the generatorh of the glauber dynamics for a model of planar rotators on a lattice in the case of a high temperature 1 we construct two so called one particle subspacesh forh and describe the spectrum of the generator in these subspaces as a consequence we find time asymptotics of the correlations for the glauber dynamics

approach to equilibrium of glauber dynamics in the one phase region i the attractive case various finite volume mixing conditions in classical statistical mechanics are reviewed and critically analyzed in particular somefinite size conditions are discussed together with their implications for the gibbs measures and for the approach to equilibrium of glauber dynamics inarbitrarily large volumes it is shown that dobrushin shlosman s theory ofcomplete analyticity and its dynamical counterpart due to stroock and zegarlinski cannot be applied in general to the whole one phase region since it requires mixing properties for regions ofarbitrary shape an alternative approach based on previous ideas of oliveri and picco is developed which allows to establish results on rapid approach to equilibrium deeply inside the one phase region in particular in the ferromagnetic case we considerably improve some previous results by holley and aizenman and holley our results are optimal in the sene that for example they show for the first time fast convergence of the dynamicsfor any temperature above the critical one for thed dimensional ising model with or without an external field in part ii we extensively consider the general case not necessarily attractive and we develop a new method based on renormalizations group ideas and on an assumption of strong mixing in a finite cube to prove hypercontractivity of the markov semigroup of the glauber dynamics

homology of the zero set of a nilpotent vector field on a flag manifold 0 1 let x be a linear transformation of a finite dimensional vector space v the configuration of flags in v which are fixed by x has rather remarkable properties when x is unipotent though this case is especially interesting the proper generality in which to study such configurations is in the theory of reductive algebraic groups where their definition can be reformulated in the language of borel subalgebras as follows let g be a connected reductive group over c with lie algebra g and let n e g be a nilpotent element let q be the variety of all borel subalgebras of g and let

on finite subgroups of su 2 simple lie algebras and the mckay correspondence the question is considered of how the restriction n where n is irreducible for all n and is a finite subgroup of su 2 decomposes into irreducibles for any arbitrary n unk it is announced that using the mckay correspondence the problem has an elegant solution in terms of the coxeter element for the associated lie algebra and that the numbers involved come in a beautiful way from the root structure

the hesselink stratification of nullcones and base change let g be a connected reductive algebraic group over an algebraically closed field of characteristic p 0 we give a case free proof of lusztig s conjectures lusztig in transform groups 10 449 487 2005 on so called unipotent pieces this presents a uniform picture of the unipotent elements of g which can be viewed as an extension of the dynkin kostant theory but is valid without restriction on p we also obtain analogous results for the adjoint action of g on its lie algebra mathfrak g and the coadjoint action of g on mathfrak g

a note on affine toda couplings the author proves the clebsch gordan property for the couplings of an untwisted affine toda field theory utilizing dorey s 1991 conjecture

two dimensional generalized toda lattice the zero curvature representation is obtained for the two dimensional generalized toda lattices connected with semisimple lie algebras the reduction group and conservation laws are found and the mass spectrum is calculated

uniqueness of the stochastic dynamics for continuous spin systems on a lattice abstract we prove the uniqueness of the stochastic dynamics associated with gibbs measures on infinite products of compact riemannian manifolds

generalised burnside rings g categories and module categories this note describes an application of the theory of generalised burnside rings to algebraic representation theory tables of marks are given explicitly for the groups s4 and s5 which are of particular interest in the context of reductive algebraic groups as an application the base sets for the nilpotent element f4 a3 are computed

killing and the coxeter transformation of kac moody algebras on etudie le spectre de la transformation de coxeter associee a une algebre de kac moody arbitraire et plus particulierement a une algebre affine

ornstein zernike asymptotics for a general two particle lattice operator we study the asymptotic behavior of correlations for a general two particle operator mathcal t acting on the hilbert space ell 2 mathbb z d times mathbb z d for all dimension d 1 2 mathcal t is written as the sum of a main term and a small interacting term a form which appears in many problems if the interacting term is small we give a complete description of the asymptotics for large t of the correlations mathcal t t f 1 f 2 t 1 2 ldots for f 1 f 2 in some suitable class the asymptotics is of the ornstein zernike type i e exponential with a power law factor which is t d for d 3 but for d 1 2 it can be anomalous and is determined by the interacting term

conjugacy classes in unitriangular matrices let gn gn q be the group of the upper unitriangular matrices of size n n over fq the finite field with q pt elements higman has conjectured that for each n the number of conjugacy classes of elements of gn is a polynomial expression in q in this paper we develop the algorithm given in j algebra 177 1995 899 introducing new ideas and theoretical properties which lead us to get the conjugacy vector of gn for n 13 these vectors for n 5 resp for n 6 7 8 were given in j algebra 152 1 1992 1 resp j algebra 177 1995 899 in particular we conclude that for these values the number of conjugacy classes of gn is a polynomial in q with integral coefficients and degree n n 6 12 thus higman s conjecture holds for n 13 on the other hand for each positive integer n we find explicitly a set of q n n 6 12 different conjugacy classes

relative entropy and mixing properties of infinite dimensional diffusions let be a diffusion process taking values on the infinite dimensional space t z where t is the circle and with components satisfying the equations d i i dw i b i dt for some coefficients i and b i i z suppose we have an initial distribution and a sequence of times t n such that lim n s tn exists where s t is the semi group of the process we prove that if i and b i are bounded of finite range have uniformly bounded second order partial derivatives and inf i i 0 then is invariant

the threshold effects for the two particle hamiltonians on lattices for a wide class of two body energy operators h k on the d dimensional lattice d d 3 k being the two particle quasi momentum we prove that if the following two assumptions i and ii are satisfied then for all nontrivial values k k 0 the discrete spectrum of h k below its threshold is non empty the assumptions are i the two particle hamiltonian h 0 corresponding to the zero value of the quasi momentum has either an eigenvalue or a virtual level at the bottom of its essential spectrum and ii the one particle free hamiltonians in the coordinate representation generate positivity preserving semi groups

nonlocal interactions and the excitation spectrum in lattice quantum scalar field models we determine the effects of nonlocal nonlinear interactions on the excitation spectrum of lattice quantum field scalar models we consider perturbations of a quantized discrete string formally self adjoint hamiltonian operator on the lattice d and with a large mass coefficient for the quadratic term the low lying energy momentum spectrum has an isolated dispersion curve and a two particle first band we analyse a ladder approximation of the bethe salpeter equation on the lattice for a weak perturbation of the type d 6 6 v 6 0 and consider the spectral interval starting at zero and extending to near the three particle threshold for space dimension d 1 2 and v 1 4 we find that a bound state occurs either below if 1 0 but not both this agrees with recent results where bound states were obtained for the stochastic dynamics generator associated with the relaxation rate to equilibrium in weakly coupled stochastic ginzburg landau models with continuous time and on a spatial lattice d these results are in contrast however with those obtained for v 2 3 for this case surprisingly we show that stable particles exist simultaneously above and below the band for d 1 2 regardless of the sign of the coupling 2 if v 3 2 2 the ladder analysis is inconclusive if d 3 4 no bound states exist in the spectral region we consider

on the spectrum of an hamiltonian in fock space discrete spectrum asymptotics a model operator h associated with the energy operator of a system describing three particles in interaction without conservation of the number of particles is considered the location of the essential spectrum of h is described the existence of infinitely many eigenvalues resp the finiteness of eigenvalues below the bottom ess h of the essential spectrum of h is proved for the case where the associated friedrichs model has a threshold energy resonance resp a threshold eigenvalue for the number n z of eigenvalues of h lying below z ess h the following asymptotics is found lim limits z to tau rm ess h 0 frac n z log z tau rm ess h u 0 0 u 0 infty

spectral analysis of a stochastic ising model in continuum we consider an equilibrium stochastic dynamics of spatial spin systems in r d involving both a birth and death dynamics and a spin flip dynamics as well using a gen eral approach to the spectral analysis of corresponding markov generator we estimate the spectral gap and construct one particle invariant subspaces for the generator

spectral analysis of weakly coupled stochastic lattice ginzburg landau models we consider the relaxation to equilibrium of solutions t 0 of stochastic dynamical langevin equations with white noise and weakly coupled ginzburg landau interactions using a feynman kac formula which relates stochastic expectations to correlation functions of a spatially non local imaginary time quantum field theory we obtain results on the joint spectrum of h where h is the self adjoint positive generator of the semi group associated with the dynamics and p j j 1 d are the self adjoint generators of the group of lattice spatial translations we show that the low lying energy momentum spectrum consists of an isolated one particle dispersion curve and for the mass spectrum energy momentum at zero momentum besides this isolated one particle mass we show using a bethe salpeter equation the existence of an isolated two particle bound state if the coefficient of the quartic term in the polynomial of the ginzburg landau interaction is negative and d 1 2 otherwise there is no two particle bound state asymptotic values for the masses are obtained

poincare series for tensor invariants and the mckay correspondence abstract for a finite group g and a finite dimensional g module v we prove a general result on the poincare series for the g invariants in the tensor algebra t v k 0 v k we apply this result to the finite subgroups g of the 2 2 special unitary matrices and their natural module v of 2 1 column vectors because these subgroups are in one to one correspondence with the simply laced affine dynkin diagrams by the mckay correspondence the poincare series obtained are the generating functions for the number of walks on the simply laced affine dynkin diagrams

stochastic models of interacting systems interacting particle systems is by now a mature area of probability theory but one that is still very active we begin this paper by explaining how models from this area arise in fields such as physics and biology we turn then to a discussion of both older and more recent results about them concentrating on contact processes voter models and exclusion processes these processes are among the most studied in the field and have the virtue of relative simplicity in their description which permits us to address the fundamental issues about their behavior without dealing with the extra complications that models from specific areas of application would require

on the intersections of a1 subgroups in the exceptional simple lie groups e6 e7 and e8 the authors determine reductions of irreducible representations of e6 e7 and e8 according to the chain of subgroups eka1tk k 6 7 8 for all a1 subgroups of ek here a1 is either su 2 or o 3 and t6 t7 t8 are respectively the tetrahedral octahedral and icosahedral groups the result is the list of coinciding branching rules for ektk proceeding via different subgroups a1

the threshold effects for a family of friedrichs models under rank one perturbations a family of friedrichs models under rank one perturbations h p p 3 0 associated to a system of two particles on the three dimensional lattice z3 is considered we prove the existence of a unique eigenvalue below the bottom of the essential spectrum of h p for all non trivial values of p under the assumption that h 0 has either a threshold energy resonance virtual level or a threshold eigenvalue the threshold energy expansion for the fredholm determinant associated to a family of friedrichs models is also obtained

on the spectrum of the generator of an infinite system of interacting diffusions we study the spectrum of the operator generating an infinite dimensional diffusion process t in space here is a natural t invariant measure on which is a gibbs distribution corresponding to a formal hamiltonian h of an anharmonic crystal with a value of the inverse temperature 0 for small enough we establish the existence of an l invariant subspace such that has a distinctive character related to a quasi particle picture in particular has a lebesgue spectrum separated from the rest of the spectrum of l and concentrated near a point 1 0 giving the smallest non zero eigenvalue of a limiting problem associated with 0

interband spectrum of weakly coupled stochastic lattice ginzburg landau models we analyze the excitation spectrum of the generator associated with the relaxation rate to equilibrium in weakly coupled stochastic ginzburg landau models on a spatial lattice z d the spectrum has a quasiparticle interpretation depending on d and on the specific interaction by solving the bethe salpeter equation in the ladder approximation we show the existence of a stable particle above the upper envelope of the two particle band possessing a concave dispersion curve this result furthers our knowledge about the spectrum of the stochastic dynamics generator

the log sobolev inequality for weakly coupled lattice fields we consider a ferromagnetic spin system with unbounded interactions on the d dimensional integer lattice d 1 under mild assumptions on the one body interactions so that arbitrarily deep double wells are allowed we prove that if the coupling constants are small enough then the finite volume gibbs states satisfy the log sobolev inequality uniformly in the volume and the boundary condition

quantum field theories in one and two dimensions axiomes pour les theories de champ quantiques theories quantiques du champ a une dimension theories quantiques du champ chirales a 2 dimensions

antilinear deformations of coxeter groups an application to calogero models we construct complex root spaces remaining invariant under antilinear involutions related to all coxeter groups we provide two alternative constructions one is based on deformations of factors of the coxeter element and the other based on the deformation of the longest element of the coxeter group motivated by the fact that non hermitian hamiltonians admitting an antilinear symmetry may be used to define consistent quantum mechanical systems with real discrete energy spectra we subsequently employ our constructions to formulate deformations of coxeter models remaining invariant under these extended coxeter groups we provide explicit and generic solutions to the schrodinger equation of these models for the eigenenergies and corresponding wavefunctions a new feature of these novel models is that when compared with the undeformed case their solutions are usually no longer singular for an exchange of an amount of particles less than the dimension of the representation space of the roots the simultaneous scattering of all particles in the model leads to anyonic exchange factors for processes which have no analogue in the undeformed case

nilpotent orbits in good characteristic and the kempf rousseau theory let g be a connected reductive algebraic group over an algebraically closed field k of characteristic p 0 image and suppose that p is a good prime for the root system of g in this paper we give a fairly short conceptual proof of pommerening s theorem pommerening j algebra 49 1977 525 536 j algebra 65 1980 373 398 which states that any nilpotent element in image is richardson in a distinguished parabolic subalgebra of the lie algebra of a levi subgroup of g as a by product we obtain a short noncomputational proof of the existence theorem for good transverse slices to the nilpotent g orbits in image for earlier proofs of this theorem see kawanaka invent math 84 1986 575 616 premet trans amer math soc 347 1995 2961 2988 spaltenstein j fac sci univ tokyo sect ia math 31 1984 283 286 we extend recent results of sommers internal math res notices 11 1998 539 562 to reductive lie algebras of good characteristic thus providing a satisfactory approach to computing the component groups of the centralisers of nilpotent elements in image and unipotent elements in g earlier computations of these groups in positive characteristics relied mostly on work of mizuno j fac sci univ tokyo sect ia math 24 1977 525 563 tokyo j math 3 1980 391 459 our approach is based on the theory of optimal parabolic subgroups for g unstable vectors also known as the kempf rousseau theory which provides a good substitute for the image theory prominent in the characteristic zero case

generalized gelfand graev representations of exceptional simple algebraic groups over a finite field i introduction let ffi be an adjoint simple algebraic group of type e n 6 7 8 f 4 or g2 over an algebraically closed field k of characteristic p 0 we assume that 5 has a fixed fq rational structure for a finite subfield fq of k the purpose of this paper is to determine explicitly the values of the irreducible characters of g ffi fq at the unipotent elements when p is good 38 i 4 3 in this part i we shall do this only for large p and the full result will be proved in the forthcoming part ii after submitting the first version of the present paper the author was informed by lusztig that he solved analogous problems for classical groups using his theory of character sheaves and that he can also recover the result of this paper mentioned above our method depends upon 1 a parametrization of the irreducible characters of g and the determination of their multiplicities in deligne lusztig virtual representations lusztig 25 2 the computation of the green functions of g in large characteristic springer 36 7 1 shoji 33 beynon and spaltenstein 3 4 3 the determination of the values of generalized gelfand graev characters 20 of g in good characteristic see section 3 by 1 and 2 the uniform parts of the irreducible characters are already known explicitly in large characteristic this implies that in large characteristic our problem for groups of type e 6 o r e 7 has already been solved it should be mentioned here that the whole character table of the g2 group is also known by chang and ree 8 hence in the g2 case this paper gives just another approach to a part of the result of chang and ree if g is of type g2 f4 or e 8 the restriction of an irreducible character r of g to the set of unipotent elements can be written as a sum of its uniform part with a function of the form

two particle spectrum of the generator for stochastic model of planar rotators at high temperatures we study two particle spectrum branches of the generator in the stochastic model of planar rotators using the construction of a special basis in two particle invariant subspaces we prove that the branches of the spectrum are in a small neighborhood of the point 2 we prove the existence of two bound states in addition to the continuous part of the spectrum in the one dimensional case

poincare series exponents of affine lie algebras and mckay slodowy correspondence abstract let n be a normal subgroup of a finite group g and v a fixed finite dimensional g module the poincare series for the multiplicities of induced modules and restriction modules in the tensor algebra t v k 0 v k are studied in connection with the mckay slodowy correspondence in particular it is shown that the closed formulas for the poincare series associated with the distinguished pairs of subgroups of su 2 give rise to the exponents of all untwisted and twisted affine lie algebras except a 2 n 1

rational points on generalized flag varieties and unipotent conjugacy in finite groups of lie type let g be a connected reductive algebraic group defined over the finite field fq where q is a power of a good prime for g we write f for the frobenius morphism of g corresponding to the fq structure so that gf is a finite group of lie type let p be an f stable parabolic subgroup of g and u the unipotent radical of p in this paper we prove that the number of uf conjugacy classes in gf is given by a polynomial in q under the assumption that the centre of g is connected this answers a question of j alperin in 1 in order to prove the result mentioned above we consider for unipotent u gf the variety p0u of g conjugates of p whose unipotent radical contains u we prove that the number of fq rational points of p0u is given by a polynomial in q with integer coefficients moreover in case g is split over fq and u is split in the sense of 22 5 the coefficients of this polynomial are given by the betti numbers of p0u we also prove the analogous results for the variety pu consisting of conjugates of p that contain u

calculating conjugacy classes in sylow p subgroups of finite chevalley groups abstract in s m goodwin on the conjugacy classes in maximal unipotent subgroups of simple algebraic groups transform groups 11 1 2006 51 76 8 the first author outlined an algorithm for calculating a parametrization of the conjugacy classes in a sylow p subgroup u q of a finite chevalley group g q valid when q is a power of a good prime for g q in this paper we develop this algorithm and discuss an implementation in the computer algebra language gap using the resulting computer program we are able to calculate the parametrization of the conjugacy classes in u q when g q is of rank at most 6 in these cases we observe that the number of conjugacy classes of u q is given by a polynomial in q with integer coefficients