kronecker products of representations of exceptional lie groups a technique is described for evaluating kronecker products of irreducible representations of an exceptional lie group g the method depends upon the natural embedding in g of a classical lie subgroup h the problem is reduced to that of finding the branching of one irreducible representation of g on restriction to h evaluating kronecker products in h and finally using the modification rules appropriate to g

bases for some reciprocity algebras ii abstract we construct bases for the stable branching algebras for the symmetric pairs gl n o n o n m o n o m and sp 2 n gl n

branching index sum rules for simple lie algebras let l and l0 be a simple lie algebra and its sub lie algebra respectively then a given irreducible representation of l decomposes into a direct sum of irreducible components of l0 which is called the branching rule the general dynkin indices introduced earlier satisfy many sum rules for the branching rule these are found to be strong enough to uniquely determine the branching rule for many cases we have studied the sum rules are especially useful for cases of exceptional lie algebras

analogies between finite dimensional irreducible representations of so 2n and infinite dimensional irreducible representations of sp 2n r i characters and products the analogy between the finite dimensional spin representation of so 2n and the infinite dimensional representation of sp 2n r is made precise it is then shown that this analogy can be extended so as to provide a precise link between each finite dimensional unitary irreducible representation of so 2n and a corresponding infinite dimensional unitary irreducible representation of sp 2n r the analogy shows itself at the level of the corresponding characters and difference characters and involves the use of schur function methods to express both characters and difference characters of so 2n and sp 2n r in terms of characters of irreducible representations of their common subgroup u n the analogy is extended still further to cover the explicit decomposition of not only tensor products of and with other unitary irreducible representations of so 2n and sp 2n r respectively but also arbitrary tensor powers of and

inverse branching rules a method is given for inverting branching rules which is based on a technique for computing kronecker products due to king 1981 this approach simplifies the calculation of plethysms and branching rules of representations of exceptional groups it may also be applied to basic spin plethysms

thermodynamic versus statistical nonequivalence of ensembles for the mean field blume emery griffiths model abstract we illustrate a novel characterization of nonequivalent statistical mechanical ensembles using the mean field blume emery griffiths beg model as a test model the novel characterization takes effect at the level of the microcanonical and canonical equilibrium distributions of states for this reason it may be viewed as a statistical characterization of nonequivalent ensembles which extends and complements the common thermodynamic characterization of nonequivalent ensembles based on nonconcave anomalies of the microcanonical entropy by computing numerically both the microcanonical and canonical sets of equilibrium distributions of states of the beg model we show that for values of the mean energy where the microcanonical entropy is nonconcave the microcanonical distributions of states are nowhere realized in the canonical ensemble moreover we show that for values of the mean energy where the microcanonical entropy is strictly concave the equilibrium microcanonical distributions of states can be put in one to one correspondence with equivalent canonical equilibrium distributions of states our numerical computations illustrate general results relating thermodynamic and statistical equivalence and nonequivalence of ensembles proved by ellis et al j stat phys 101 2000 999

young tableau methods for kronecker products of representations of the classical groups diagrammatic methods for decomposing kronecker products of arbitrary representations of any of the classical groups are presented for convenience efficient ways of computing the dimensions and quadratic casimir s c2 r are also given these methods seem more useful for hand calculations than the method of schur functions or characteristic polynomials an appendix presents the kronecker products for any two representations of dimension 100

complete analysis of phase transitions and ensemble equivalence for the curie weiss potts model using the theory of large deviations we analyze the phase transition structure of the curie weiss potts spin model which is a mean field approximation to the nearest neighbor potts model it is equivalent to the potts model on the complete graph on n vertices the analysis is carried out both for the canonical ensemble and the microcanonical ensemble besides giving explicit formulas for the microcanonical entropy and for the equilibrium macrostates with respect to the two ensembles we analyze ensemble equivalence and nonequivalence at the level of equilibrium macrostates relating these to concavity and support properties of the microcanonical entropy the curie weiss potts model is the first statistical mechanical model for which such a detailed and rigorous analysis has been carried out

statistical mechanics in the extended gaussian ensemble the extended gaussian ensemble ege is introduced as a generalization of the canonical ensemble this ensemble is a further extension of the gaussian ensemble introduced by hetherington j low temp phys 66 145 1987 the statistical mechanical formalism is derived both from the analysis of the system attached to a finite reservoir and from the maximum statistical entropy principle the probability of each microstate depends on two parameters b and g which allow one to fix independently the mean energy of the system and the energy fluctuations respectively we establish the legendre transform structure for the generalized ther modynamic potential and propose a stability criterion we also compare the ege probability distribution with the q exponential distribution as an example an application to a system with few independent spins is presented the development of statistical mechanics based on en semble theory is founded on the postulate of equal a priori probabilities which is assumed to apply to all microstates consistent with the given macrostate of an isolated system 1 the corresponding statistical ensemble is the so called microcanonical ensemble a representative system in this en semble has all mechanical variables such as energy e volume v magnetization m etc fixed for convenience in calculations other ensembles are used which invariably sup pose the existence of a subsidiary system or reservoir in con tact with the actual system for instance in the canonical ensemble the walls of the system permit an exchange of en ergy with the reservoir while in the grand canonical en semble both energy and matter can be exchanged in gen eral the different ensembles are constructed by allowing one or more mechanical variables to fluctuate the exchange of each of these variables is controlled by a parameter which is a characteristic of the reservoir for instance in the case of the canonical ensemble this parameter is precisely the tem perature of the reservoir and determines the mean energy of the system actually this is adequate when the reservoir is a very large system that can exchange arbitrary amounts of energy without modification of its intensive properties in practical situations this is not always the case however very few studies have been devoted to analyze the conse quences of possible deviations from these ideal reservoir properties in this paper we develop the statistical mechanics of a system that can exchange energy with a finite reservoir char acterized by two parameters b and g these parameters con trol independently the mean energy of the system and its energy fluctuations respectively the corresponding statisti cal ensemble represents a generalization of the canonical en semble and will be called the extended gaussian ensemble ege a similar ensemble was already developed in a more restricted framework by hetherington 2 the author con sidered that the sample system was in contact with a finite reservoir with size dependent properties the so called gaussian ensemble was introduced so that it is equivalent to the canonical ensemble in the limit of large systems except in the energy range of a first order transition interestingly it enables a smooth interpolation between the microcanonical and the canonical ensembles taking into account these fea tures challa and hetherington 3 4 showed the interest of this ensemble for monte carlo simulation studies of phase transitions they demonstrated a significant reduction in computer time compared to standard simulations in the ca nonical ensemble and its adequacy for distinguishing second order from first order transitions compared to the ege introduced in the present paper the main difference arises from the fact that in the gaussian ensemble the sample and the reservoir are assumed to be statistically independent which implies the additivity of the corresponding entropies this is not assumed in our formalism the consequences are important and will be discussed in depth in this work the present formalism can be considered as an alternative to the statistical mechanics based on nonadditive generalized entropies actually the study of such generalized entropies has generated a lot of interest in the past 15 years the mo tivation for the so called tsallis statistical mechanics has been to extend the standard boltzmann gibbs framework to include nonextensive systems 5 among different interpre tations it has been suggested that tsallis formalism corre sponds to an ensemble describing a system attached to a finite reservoir 6 7 although a large number of papers have been published the physical meaning of many related issues is still open to discussion 8 9 the ege formalism that we propose in this paper provides a clear and consistent frame work for the statistical mechanics with a finite reservoir the paper is organized as follows in secs ii and iii the ege is founded from the analysis of a contact with a finite reservoir and from the maximum statistical entropy prin ciple respectively in sec iv the main thermodynamic rela tions are derived in sec v we highlight the nonadditive nature of the thermodynamic formalism in sec vi a stabil

an explicit equivalent positive semidefinite program for nonlinear 0 1 programs we consider the general nonlinear optimization problem in 0 1 variables and provide an explicit equivalent positive semidefinite program in 2n 1 variables the optimal values of both problems are identical from every optimal solution of the former one easily finds an optimal solution of the latter and conversely from every solution of the latter one may construct an optimal solution of the former for illustration the equivalent positive semidefinite program is explicated for quadratic 0 1 programs and max cut in mathbb r 3 for unconstrained 0 1 programs a special representation in terms of a weighted sum of squares is provided

composite young diagrams supercharacters of u m n and modification rules the supercharacter of u m n associated with an arbitrary composite young diagram is defined the distinction is made between standard and non standard supercharacters a modification rule is presented which may be used to express any non standard supercharacter in terms of standard supercharacters in the case n 0 the rule reduces to the rule already known to be appropriate to u m

algebraic solution for the vector potential in the dirac equation the dirac equation for an electron in an external electromagnetic field can be regarded as a singular set of linear equations for the vector potential radford s method of algebraically solving for the vector potential is reviewed with attention to the additional constraints arising from non maximality of the rank the extension of the method to general spacetimes is illustrated by examples in diverse dimensions with both c and a number wavefunctions pacs numbers 0350d 0210 0220 0365p 1130p

a comparison of the sherali adams lov 341sz schrijver and lasserre relaxations for 0 1 programming sherali and adams sa90 lov asz and schrijver ls91 and recently lasserre las01b have proposed lift and project methods for constructing hierarchies of successive linear or semidefinite relaxations of a 0 1 polytope p subseteq or n converging to p in n steps lasserre s approach uses results about representations of positive polynomials as sums of squares and the dual theory of moments we present the three methods in a common elementary framework and show that the lasserre construction provides the tightest relaxations of p as an application this gives a direct simple proof for the convergence of the lasserre s hierarchy we describe applications to the stable set polytope and to the cut polytope

holomorphic discrete series and harmonic series unitary irreducible representations of non compact lie groups sp 2n r u p q and so 2n generalised characters of the infinite dimensional holomorphic discrete series unitary irreducible representations of the non compact groups u p q sp 2n r and so 2n are explicitly expressed in terms of characters of finite dimensional unitary group representations these formulae are remarkably succinct despite involving certain infinite series of schur functions similar formulae are derived for harmonic series unitary representation of both u p q and sp 2n r consideration of the branching rules from u p q to u q p and from sp 2n r to u n enables holomorphic representations to be identified as a subset of the harmonic representations the branching rules are established in full generality and are then used in the evaluation of tensor products of both holomorphic and harmonic representations in the case of the former a known result is recast in terms of closed formulae involving schur functions and for the latter various generalisations of these formulae are given a conjecture is also made regarding what might be the simplest possible formulae covering all holomorphic and harmonic representations of sp 2n r and u p q illustrative examples are presented

s function series revisited a systematic and readily programmable computational method is given to extract the s function content of any arbitrary symmetric polynomial function applications to the derivation of new series and the outer product or skew division of an s function by a series are shown

a general representation theorem for partially ordered commutative rings an extension of the kadison dubois representation theorem is proved this extends both the classical version 3 and the preordering version given by jacobi in 5 it is then shown how this can be used to sharpen the results on representations of strictly positive polynomials given by jacobi and prestel in 6

young diagrams supercharacters of osp m n and modification rules the supercharacter of osp m n associated with an arbitrary young diagram is defined the distinction is made between osp m n standard and non standard supercharacters the corresponding modification rule which may be used to express a non standard supercharacter in terms of standard supercharacters is presented exemplified and proved this rule involves the removal of n 2 1 boundary strips from the young diagram in the case n 0 the rule reduces to the well known rule appropriate to o m for a non standard supercharacter corresponding to a typical irreducible representation of osp m n the modification yields a single typical standard supercharacter on the other hand for a non standard supercharacter corresponding to an atypical irreducible representation the rule yields a linear combination of atypical standard supercharacters

extended poincare supersymmetry rotation groups and branching rules the decomposition of the basic spin irreps of the special orthogonal group so2k into irreps of sod 2 k where d is the spacetime dimension of the extended d dimensional poincare supersymmetry and k is the appropriate automorphism group is determined general results for d 10 several new branching rules for subgroups of so2k are developed a number of specific results are tabulated

optimization of polynomials on compact semialgebraic sets a basic closed semialgebraic subset s of r n is defined by simultaneous polynomial inequalities g 1 ge 0 dotsc g m ge 0 we give a short introduction to lasserre s method for minimizing a polynomial f on a compact set s of this kind it consists of successively solving tighter and tighter convex relaxations of this problem which can be formulated as semidefinite programs we give a new short proof for the convergence of the optimal values of these relaxations to the infimum f ast of f on s which is constructive and elementary in the case where f possesses a unique minimizer x ast we prove that every sequence of nearly optimal solutions of the successive relaxations gives rise to a sequence of points in r n converging to x ast

a lift and project cutting plane algorithm for mixed 0 1 programs we propose a cutting plane algorithm for mixed 0 1 programs based on a family of polyhedra which strengthen the usual lp relaxation we show how to generate a facet of a polyhedron in this family which is most violated by the current fractional point this cut is found through the solution of a linear program that has about twice the size of the usual lp relaxation a lifting step is used to reduce the size of the lp s needed to generate the cuts an additional strengthening step suggested by balas and jeroslow is then applied we report our computational experience with a preliminary version of the algorithm this approach is related to the work of balas on disjunctive programming the matrix cone relaxations of lovasz and schrijver and the hierarchy of relaxations of sherali and adams

spectrum generating functions for strings and superstrings the spectrum generating functions for a variety of strings and superstrings are derived these are expressed in terms of characters of irreducible representations of the appropriate transverse spacetime symmetry group and in the case of heterotic strings the relevant gauge group e8 e8 or spin 32 z2 the expansions involve principal specialisations of characters of both gl n and gl m n in the limit as m and n tend to infinity by using schur function methods modification rules and various theta function identities a number of spectrum generating functions are derived and tabulated the equivalence of the gso projected neveu schwarz ramond superstring and the green schwarz superstring is proved and the connection between the lattice formulation and the gso projected neveu schwarz ramond formulation of heterotic strings is spelled out

a sum of squares approximation of nonnegative polynomials we show that every real nonnegative polynomial f can be approximated as closely as desired in the l 1 norm of its coefficient vector by a sequence of polynomials f epsilon that are sums of squares the novelty is that each f epsilon has a simple and explicit form in terms of f and epsilon

symmetrised powers of rotation group representations the results given permit the unambiguous evaluation of all possible kronecker products of the irreducible representations tensor and spinor of on and son for n 2 nu and n 2 nu 1 a complete resolution of the second and third powers of the basic spinor representations of so2 nu and so2 nu 1 is given together with a prescription for analysing the fourth power of these representations detailed application is made to the enumeration of properties of so10 relevant to grand unified theories and sufficient information given to resolve the fourth power of any representation of so10

upper bounds for multiplicities in a tensor product of representations and in a restriction of group representations to a subgroup upper bounds for multiplicities of irreducible representations of a semisimple compact lie group in the tensor product of representations and multiplicities of irreducible representations of a subgroup in an irreducible representation of a group are derived pairs of groups and subgroups appear from the representation theory of non compact groups

cones of matrices and set functions and 0 1 optimization it has been recognized recently that to represent a polyhedron as the projection of a higher dimensional but simpler polyhedron is a powerful tool in polyhedral combinatorics a general method is developed to construct higher dimensional polyhedra or in some cases convex sets whose projection approximates the convex hull of 0 1 valued solutions of a system of linear inequalities an important feature of these approximations is that one can optimize any linear objective function over them in polynomial time in the special case of the vertex packing polytope a sequence of systems of inequalities is obtained such that the first system already includes clique odd hole odd antihole wheel and orthogonality constraints in particular for perfect and many other graphs this first system gives the vertex packing polytope for various classes of graphs including t perfect graphs it follows that the stable set polytope is the projection of a polytope with a polynomial number of facets an extension o

vertex operators and composite supersymmetric s functions the s function basis for the fock space of vertex operator constructions is considered where states are associated with partitions mod lambda the authors show that the matrix element lambda mod valpha z mod mu is essentially given by the composite s function slambda mu z where the argument z identical to z 1 is replicated mod alpha mod times matrix elements of products of vertex operators lambda mod valpha z v beta w mod mu are given up to functions of z w independent of lambda and mu by the composite supersymmetric s function slambda mu z w where the z w are replicated mod alpha mod mod beta mod times the supersymmetric arguments being those associated with negative roots

symmetric and unitary group representations i duality theory an extension of the schur weyl duality connecting the representations of the symmetric and unitary groups is given the schur weyl basis is constructed using annihilation and creation operators three factorisation lemmas are derived their importance lies in the fact that they relate the phase freedoms within the racah wigner algebra of the symmetric groups and the unitary groups extensions of the regge symmetries are also given these are expressed in five duality relations

new example of a generating function for wznw fusion rules the authors review the construction of generating functions for wznw wess zumino novikov witten fusion rules and present the explicit form of the generating function for so 5 in addition to providing a compact and manifestly positive expression for the so 5 fusion rules at all levels this result illustrates a general conjecture on the relation between the generating function for wznw fusion rules and that for finite lie algebra tensor products

the representation space of the nuclear symplectic sp 6 r shell model the problem of obtaining the sp 6 r irreducible representations that arise from the symmetrized products of a copies of the fundamental representation of sp 6 r is considered a simple method is developed that rapidly leads to the determination of the sp 6 r content up to a prescribed cut off in practical calculations a is identified with the nucleon number it is shown that for a given excitation energy cut off n omega and sufficiently large a the number and multiplicities of the sp 6 r irreducible representations is fixed and the content can be expressed in an a independent manner complete results are given for n or 16 there is no difficulty in principle in going to higher excitations and details are given for determining the sp 6 r content for a or 16

convexity in semialgebraic geometry and polynomial optimization we review several and provide new results on the theory of moments sums of squares and basic semialgebraic sets when convexity is present in particular we show that under convexity the hierarchy of semidefinite relaxations for polynomial optimization simplifies and has finite convergence a highly desirable feature as convex problems are in principle easier to solve in addition if a basic semialgebraic set mathbf k is convex but its defining polynomials are not we provide two algebraic certificates of convexity which can be checked numerically the second is simpler and holds if a sufficient and almost necessary condition is satisfied it also provides a new condition for mathbf k to have semidefinite representation for this we use and extend some of the recent results from the author and helton and nie math program to appear finally we show that when restricting to a certain class of convex polynomials the celebrated jensen s inequality in convex analysis can be extended to linear functionals that are not necessarily probability measures

the theory of angular momenta and higher su n symmetries the authors discuss interrelations between generating invariants gi for the reduction pi mi 1 d pi to d on 1 of the m fold tensor product of su n irreducible representations irreps d pi and polynominal bases of the su m irreps d on 1j om n 1 ok 0 0 0 k times a realisation of the su m irrep d oj om 3 bases is given in terms of gi for su 2 coupling wigner coefficients as a byproduct an expression is obtained for su 2 6 j symbols in terms of only two wigner coefficients the authors also discuss some possibilities of the analysis involved in solving the wigner biedenharn problem construction of orthonormal sets of the wigner coefficients for su n groups n or 3

branching rules and even dimensional rotation groups so2k unambiguous methods are developed for calculating branching rules for the classical subgroups of the even dimensional rotation group so2k complete results are given for the subgroups suk u1 so2k 2 u1 so2p so2q and so2p 1 so2q 1 a number of examples relevant to problems in supergravity and unification theories are given a complete resolution of the antisymmetric powers of the basic spinor irrep of so10 is given and the results extended to so11

demazure characters and wzw fusion rules the demazure character formula is applied to the verlinde formula for the fusion rules of wess zumino witten wzw models we follow littelmann s derivation of a generalized littlewood richardson rule that computes tensor products for simple lie algebras a combinatorial rule for wzw fusions does not result however only a modified version of the littlewood richardson rule is obtained that computes an old upper bound on the fusion coefficients of ar wzw models we argue that this is because the characters of simple lie algebras appear in this treatment instead of the corresponding affine characters the bruhat order on the affine weyl group must be implicated in any combinatorial rule for wzw fusions the bruhat order on subgroups of this group such as the finite weyl group does not suffice

a classification of multiplicity free actions abstract the action of a complex algebraic groupgon an affine varietyvis said to be multiplicity free if each irreducible representation ofgoccurs at most once in the ring c v of regular functions onv this paper concerns the classical setting wherevis a vector space andgacts linearly onv the irreducible linear multiplicity free actions have been classified by v kac the focus here is on nonirreducible linear multiplicity free actions we exhibit many such actions which do not decompose as direct products of irreducible actions our main results provide a classification for all linear multiplicity free actions on a vector space

canonical and microcanonical partition functions in long range systems with order parameter space of arbitrary dimension abstract we consider systems in which the canonical partition function can be expressed as the integral in an n dimensional space the order parameter space of a function that also depends parametrically on the number n of degrees of freedom and on the inverse temperature we show how to compute together with the canonical entropy and specific heat also the corresponding microcanonical quantities generalizing in this way some results already in the literature for the case n 1 from the expressions that are obtained it is possible to derive the necessary conditions for the equivalence of the canonical and of the microcanonical ensembles we finally study a simple model with a two dimensional order parameter space in which ensemble inequivalence is realized

semidefinite representations for finite varieties we consider the problem of minimizing a polynomial over a set defined by polynomial equations and inequalities when the polynomial equations have a finite set of complex solutions we can reformulate this problem as a semidefinite programming problem our semidefinite representation involves combinatorial moment matrices which are matrices indexed by a basis of the quotient vector space x 1 xn i where i is the ideal generated by the polynomial equations in the problem moreover we prove the finite convergence of a hierarchy of semidefinite relaxations introduced by lasserre semidefinite approximations can be constructed by considering truncated combinatorial moment matrices rank conditions are given in a grid case that ensure that the approximation solves the original problem to optimality

branching rules for e8 so16 branching rules for e8 down arrow so16 are derived from those of e8 down arrow su2 e7 down arrow su8 and so16 down arrow u1 su8 by noting that both group chains share a common u1 su8 subgroup additional branching rules are deduced from kronecker products of e8 and so16 irreps in each case the rules distinguish unambiguously between conjugate irreps of so16 an alternative labelling scheme for the irreps of so2k based on its maximal u1 suk subgroup is outlined

unitary representations branching rules and matrix elements for the non compact symplectic groups the complementarity of the symplectic and orthogonal groups is used to infer properties of the infinite dimensional unirreps of the former from the character theory of the latter the complete set of d series metaplectic unirreps of sp n r is identified and branching rules are given for their restrictions to the maximal compact subgroup u n developed in terms of the properties of schur functions a known algorithm for the evaluation of matrix elements of the sp 3 r lie algebra is extended to any sp n r and analytic expressions are given for important classes of unirreps and multiplicity free states

plethysm for the non compact group and new s function identities methods of computing plethysms of the fundamental unitary irreducible representations of the non compact symplectic group sp 2n r are considered complete results are given for the symmetrized second powers a number of new s function identities are reported the stability properties of the sp 2n r plethysms are noted as well as a remarkable conjugacy relation the application of the plethysms to n particles in an isotropic harmonic oscillator is briefly outlined

natural labelling schemes for simple roots and irreducible representations of exceptional lie algebras the simple root system of each exceptional simple lie algebra is explicitly constructed in a variety of forms each construction is based on the natural embedding in the exceptional algebra of a classical semi simple algebra of the same rank the procedure adopted leads to the discovery of a number of new chains of subalgebras illustrated by means of supplemented dynkin diagrams the various sets of simple roots are then used to determine natural labels for the irreducible representations of each of the exceptional algebras for each such labelling scheme the modification rules for dealing with non standard representation labels are tabulated the connection between the natural labels and dynkin labels is given in detail and a comparison is made with the labels of b g wybourne and m j bowick 1977

a generalization of the littlewood richardson rule for the general linear group gz the littlewood richardson rule see 18 193 gives a method to calculate the decomposition of a tensor product of two irreducible g representations the aim of this article is to give a generalization of this rule for all simple simply connected algebraic groups of type a b c d g2 and e we obtain also partial results for g of type f4 e and es the restrictions in the last three cases come from the fact that for the formulation of the decomposition rules we need the notion of a standard young tableau such a notion has been developed by seshadri lakshmibai musili and rajeswari in a series of articles see ll 13 14 16 but not yet for all representations of the last three exceptional groups the advantage of the notion of a young tableau developed by seshadri et al is that it is independent of the type of the group for the convenience of the reader not used to this notion we give first a seperate proof for g sz using the classical notion of a standard young tableau of course for applications the classical notion is much more appropriate in the appendix we give a translation of the notion of a standard young tableau in the sense of seshadri et al into the classical notion of a young tableau for g sp2 and spin in 3 8 we give also such a translation for g g for the other exceptional groups such a translation is not known to the author let p be a dominant weight for g and denote by i the corresponding simple g module we assume that the character of vp is given by ct eycr where the sum has to be taken over all standard young tableaux f of shape p p and v 5 is the weight of the tableau y now by the

the moment problem on compact semi algebraic sets in this chapter we begin the study of the multidimensional moment problem the passage to dimensions d 2 brings new difficulties and unexpected phenomena in sect 3 2 we derived solvability criteria of the moment problem on intervals in terms of positivity conditions it seems to be natural to look for similar characterizations in higher dimensions as well this leads us immediately into the realm of real algebraic geometry and to descriptions of positive polynomials on semi algebraic sets in this chapter we treat this approach for basic closed compact semi algebraic subsets of mathbb r d it turns out that for such sets there is a close interaction between the moment problem and positivstellensatze for strictly positive polynomials

negative temperature states and large scale long lived vortices in two dimensional turbulence we study onsager s theory of large coherent vortices in turbulent flows in the approximation of the point vortex model for two dimensional euler hydrodynamics in the limit of a large number of point vortices with the energy perpair of vortices held fixed we prove that the entropy defined from the microcanonical distribution as a function of the pair specific energy has its maximum at a finite value and thereafter decreases yielding the negative temperature states predicted by onsager we furthermore show that the equilibrium vorticity distribution maximizes an appropriate entropy functional subject to the constraint of fixed energy and under regularity assumptions obeys the joyce montgomery mean field equation we also prove that under appropriate conditions the vorticity distribution is the same as that for the canonical distribution a form of equivalence of ensembles we establish a large fluctuation theory for the microcanonical distributions which is based on a level 3 large deviations theory for exchangeable distributions we discuss some implications of that property for the ergodicity requirements to justify onsager s theory and also the theoretical foundations of a recent extension to continuous vorticity fields by r robert and j miller although the theory of two dimensional vortices is of primary interest our proofs actually apply to a very general class of mean field models with long range interactions in arbitrary dimensions

kronecker products for so 2p representations a relatively simple algorithm for the decomposition of the product of two so 2p representations is presented for this purpose generalised young tableaux are introduced and their product is defined

simultaneity and reality of u n and su n 3jm and 6j symbols a simple proof is given of the simultaneous existence of a real set of 3jm and 6j symbols for the unitary groups u n and su n including the case of mixed tensor representations we observe that simultaneity is incompatible with the conventional permutation symmetries for u n with n 3 the relevance of these results to the schur weyl duality of u n with the symmetric groups sl is discussed

decomposition of the polynomial kernel of arbitrary higher spin dirac operators in a series of recent papers we have introduced higher spin dirac operators which are generalisations of the classical dirac operator whereas the latter acts on spinor valued functions the former acts on functions taking values in arbitrary irreducible half integer highest weight representations for the spin group in this paper we describe how the polynomial kernel spaces of such operators decompose in irreducible representations of the spin group we will hereby make use of results from representation theory

the weyl groups and weight multiplicities of the exceptional lie groups the weyl group wg of each exceptional simple lie group g is described in detail its structure is defined in terms of its coset decomposition with respect to the weyl group wh of a classical semi simple lie group h embedded naturally in g the concepts of g dominance and g equivalence are defined and used to determine from the character formula of weyl the branching rule associated with the restriction of group elements from g to h the weyl group wg is used further to impose constraints on both the branching multiplicities for g to h and the weight multiplicities of g these constraints are used to evaluate the weight multiplicities of f4 e6 e7 and e8 together with the branching multiplicities for e8 to so 16

new s function series and non compact lie groups a number of new infinite series of s functions are described in terms of their generating functions and s function content applications to the character theory of non compact lie groups are noted

univariate parameterization for global optimization of mixed integer polynomial problems abstract this paper presents a new relaxation technique to globally optimize mixed integer polynomial programming problems that arise in many engineering and management contexts using a bilinear term as the basic building block the underlying idea involves the discretization of one of the variables up to a chosen accuracy level teles j p castro p m matos h a 2013 multiparametric disaggregation technique for global optimization of polynomial programming problems j glob optim 55 227 251 by means of a radix based numeric representation system coupled with a residual variable to effectively make its domain continuous binary variables are added to the formulation to choose the appropriate digit for each position together with new sets of continuous variables and constraints leading to the transformation of the original mixed integer non linear problem into a larger one of the mixed integer linear programming type the new underestimation approach can be made as tight as desired and is shown capable of providing considerably better lower bounds than a widely used global optimization solver for a specific class of design problems involving bilinear terms

slinkies and the s function content of certain generating functions a variety of methods are described for expanding particular generating functions in terms of their s function content plethysms are used to establish conjugacy relations between s function series and the task of evaluating certain products of classical s function series introduced by littlewood 1950 is then completed slinky diagrams are introduced to represent non standard s functions and their modification to give standard s functions the role of slinkies in determining the s function content of other generating functions is then explained and exemplified

global optimization with polynomials and the problem of moments we consider the problem of finding the unconstrained global minimum of a real valued polynomial p x mathbb r n to mathbb r as well as the global minimum of p x in a compact set k defined by polynomial inequalities it is shown that this problem reduces to solving an often finite sequence of convex linear matrix inequality lmi problems a notion of karush kuhn tucker polynomials is introduced in a global optimality condition some illustrative examples are provided

metastability within the generalized canonical ensemble we discuss a property of our recently introduced generalized canonical ensemble m costeniuc r s ellis h touchette b turkington the generalized canonical ensemble and its universal equivalence with the microcanonical ensemble j stat phys 119 2005 1283 we show that this ensemble can be used to transform metastable or unstable nonequilibrium states of the standard canonical ensemble into stable equilibrium states within the generalized canonical ensemble equilibrium calculations within the generalized canonical ensemble can thus be used to obtain information about nonequilibrium states in the canonical ensemble r 2006 elsevier b v all rights reserved

kronecker products for compact semisimple lie groups a review is given of the application of s function techniques to the evaluation of kronecker products of irreducible representations of compact semisimple lie groups explicit formulae are derived for all irreducible representations of all such groups recent developments involving composite young diagrams are brought to fruition and the vexed problem of so2k is dealt with completely new branching rules for the classical groups are given in an appendix these are exploited in the evaluation of kronecker products by means of a technique which is applied to both classical and exceptional groups a discussion is made of various modification rules which are needed to express the final results in standard form

stable branching rules for classical symmetric pairs we approach the problem of obtaining branching rules from the point of view of dual reductive pairs specifically we obtain a stable branching rule for each of 10 classical families of symmetric pairs in each case the branching multiplicities are expressed in terms of littlewood richardson coefficients some of the formulas are classical and include for example littlewood s restriction rule as a special case

adjoint affine fusion and tadpoles we study affine fusion with the adjoint representation for simple lie algebras elementary and universal formulas determine the decomposition of a tensor product of an integrable highest weight representation with the adjoint representation using the refined affine depth rule we prove that equally striking results apply to adjoint affine fusion for diagonal fusion a coefficient equals the number of nonzero dynkin labels of the relevant affine highest weight minus 1 a nice lattice polytope interpretation follows and allows the straightforward calculation of the genus 1 1 point adjoint verlinde dimension the adjoint affine fusion tadpole explicit formulas piecewise polynomial in the level are written for the adjoint tadpoles of all classical lie algebras we show that off diagonal adjoint affine fusion is obtained from the corresponding tensor product by simply dropping non dominant representations

heisenberg weyl algebras of symmetric and antisymmetric bosons all the polynomials in symmetric 2 and antisymmetric 11 u n bosons are constructed in u n bases and u n reduced matrix elements for the bosons between polynomial basis states are computed applications to representation theory of lie groups paired fermion and boson physics are briefly discussed

covariance correlation and entanglement some new identities for quantum variance and covariance involving commutators are presented in which the density matrix and the operators are treated symmetrically a measure of entanglement is proposed for bipartite systems based upon covariance this works for two and three component systems but produces ambiguities for multicomponent systems of composite dimension its relationship to angular momentum dispersion for symmetric spin states is described

characters dimensions and branching rules for covariant irreps of u n m the theory of symmetric functions is used to obtain simple expressions for the characters and dimensions of the covariant irreps of the supergroup u n m schur function expressions are given for a number of important branching rules

new branching rules induced by plethysm we derive group branching laws for formal characters of subgroups of leaving invariant an arbitrary tensor t of young symmetry type where is an integer partition the branchings and fixing a vector vi a symmetric tensor gij gji and an antisymmetric tensor fij fji respectively are obtained as special cases all new branchings are governed by schur function series obtained from plethysms of the schur function s by the basic m series of complete symmetric functions and the l m 1 series of elementary symmetric functions our main technical tool is that of hopf algebras and our main result is the derivation of a coproduct for any schur function series obtained by plethysm from another such series therefrom one easily obtains generalized newell littlewood formulae and the algebra of the formal group characters of these subgroups is established concrete examples and extensive tabulations are displayed for and showing their involved and nontrivial representation theory the nature of the subgroups is shown to be in general affine and in some instances non reductive we discuss the complexity of the coproduct formula and give a graphical notation to cope with it we also discuss the way in which the group branching laws can be reinterpreted as twisted structures deformed by highly nontrivial 2 cocycles the algebra of subgroup characters is identified as a cliffordization of the algebra of symmetric functions for formal characters modification rules are beyond the scope of the present paper but are briefly discussed

kronecker product of sp 2n representations using generalised young tableaux using generalised young tableaux the authors obtain an explicit formula for the reduction of the kronecker product of irreducible representations of the symplectic groups this extends a previous work devoted to the case of orthogonal groups

tensor gauge fields in arbitrary representations of gl d mathbb r ii quadratic actions quadratic second order non local actions for tensor gauge fields transforming in arbitrary irreducible representations of the general linear group in d dimensional minkowski space are explicitly written in a compact form by making use of levi civita tensors the field equations derived from these actions ensure the propagation of the correct massless physical degrees of freedom and are shown to be equivalent to non lagrangian local field equations proposed previously moreover these actions allow a frame like reformulation a la macdowell mansouri without any trace constraint in the tangent indices

semidefinite programming vs lp relaxations for polynomial programming we consider the global minimization of a multivariate polynomial on a semi algebraic set o defined with polynomial inequalities we then compare two hierarchies of relaxations namely lp relaxations based on products of the original constraints in the spirit of the rlt procedure of sherali and adams 1990 and recent semidefinite programming sdp relaxations introduced by the author the comparison is analyzed in light of recent results in real algebraic geometry on various representations of polynomials positive on a compact semi algebraic set

the hopf algebra structure of the character rings of classical groups the character ring of covariant irreducible tensor representations of the general linear group admits a hopf algebra structure isomorphic to the hopf algebra of symmetric functions here we study the character rings and of the orthogonal and symplectic subgroups of the general linear group within the same framework of symmetric functions we show that and also admit natural hopf algebra structures that are isomorphic to that of and hence to the isomorphisms are determined explicitly along with the specification of standard bases for and analogous to those used for a major structural change arising from the adoption of these bases is the introduction of new orthogonal and symplectic schur hall scalar products significantly the adjoint with respect to multiplication no longer coincides as it does in the case with a foulkes derivative or skew operation the adjoint and foulkes derivative now require separate definitions and their properties are explored here in the orthogonal and symplectic cases moreover the hopf algebras and are not self dual the dual hopf algebras and are identified finally the hopf algebra of the universal rational character ring of mixed irreducible tensor representations of the general linear group is introduced and its structure maps identified

su n and sp 2n wzw fusion rules fusion rules for wzw models based on su n and sp 2n are considered using the results of kac 1989 and walton 1989 1990 it is shown that these fusion rules may be computed using young diagram methods by performing extra modifications these modifications take the form of the removal of boundary strips which bear a striking resemblance to the modifications necessary when performing tensor products for these algebras this fact is exploited to exhibit a duality between the fusion rules of su n at level k and su k at level n and also between sp 2n at level k and so 2k at level n the former duality has been discussed in the context of two dimensional statistical mechanics by kuniba and nakanishi 1990 and also by naculich and schnitzer 1990 in the case of wzw models and goodman and wenzl 1989 for hecke algebras at roots of unity for su 3 a manifestly positive combinatorial procedure for computing fusion rules and a generating function for the fusion coefficients are given

a joint marginal approach to parametric polynomial optimization given a compact parameter set mathbf y subset mathbb r p we consider polynomial optimization problems mathbf p mathbf y on mathbb r n whose description depends on the parameter mathbf y in mathbf y we assume that one can compute all moments of some probability measure varphi on mathbf y absolutely continuous with respect to the lebesgue measure e g mathbf y is a box or a simplex and varphi is uniformly distributed we then provide a hierarchy of semidefinite relaxations whose associated sequence of optimal solutions converges to the moment vector of a probability measure that encodes all information about all global optimal solutions mathbf x mathbf y of mathbf p mathbf y as mathbf y in mathbf y in particular one may approximate as closely as desired any polynomial functional of the optimal solutions like e g their varphi mean in addition using this knowledge on moments the measurable function mathbf y mapsto x k mathbf y of the k th coordinate of optimal solutions can be estimated e g by maximum entropy methods also for a boolean variable x k one may approximate as closely as desired its persistency varphi mathbf y x k mathbf y 1 i e the probability that in an optimal solution mathbf x mathbf y the coordinate x k mathbf y takes the value 1 last but not least from an optimal solution of the dual semidefinite relaxations one provides a sequence of polynomial resp piecewise polynomial lower approximations with l 1 varphi resp varphi almost uniform convergence to the optimal value function

polynomial programming lp relaxations also converge we consider the global minimization of a multivariate polynomial on a compact semialgebraic set using a result of krivine j analyse math 12 1964 pp 307 326 c r acad sci paris 258 1964 pp 3417 3418 it is shown that lp relaxations based on products of the original constraints in the spirit of the reformulation linearization technique procedure of sherali and adams converge to the global optimum this extends to compact semialgebraic sets previous results known for polytopes and 0 1 programs the asymptotic behavior of the lp relaxations is also analyzed and it is shown that in important cases finite convergence cannot occur

a hopf laboratory for symmetric functions an analysis of symmetric function theory is given from the perspective of the underlying hopf and bi algebraic structures these are presented explicitly in terms of standard symmetric function operations particular attention is focused on laplace pairing sweedler cohomology for 1 and 2 cochains and twisted products rota cliffordizations induced by branching operators in the symmetric function context the latter are shown to include the algebras of symmetric functions of orthogonal and symplectic type a commentary on related issues in the combinatorial approach to quantum field theory is given