reduced notation inner plethysms and the symmetric group the reduced notation for irreducible representations of the symmetric group s is interpreted in terms of symmetric formal series and vertex operators and is used to prove a number of properties of reduced kronecker products and inner plethysms in an n independent manner conditions for self associativity of kronecker products and inner plethysms are established reduced inner plethysms are developed and applied to the question of non simple phase groups among the symmetric sn and alternating an groups

vector coherent state theory of the noncompact orthosymplectic superalgebras ii some selected examples for pt i see ibid vol 23 p 5383 1990 the techniques presented in the first paper of the present series for determining the conditions for the existence of star or grade star positive discrete series irreducible representations of osp p 2n r p 2m or 2m 1 and the branching rule for their decomposition into a direct sum of so p sp 2n r irreducible representations as well as for constructing explicit matrix realizations are illustrated with a few selected examples the latter include the most general irreps of osp 1 2n 2 osp 2 2 r osp 3 2 r osp 4 2 r osp 2 4 r and the most degenerate irreps of osp 2 2n r in addition all the information necessary for dealing with other cases amenable to a full analytic treatment is provided

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

a filled function method for finding a global minimizer on global integer optimization the paper gives a definition of the filled function for nonlinear integer programming this definition is modified from that of the global convexized filled function for continuous global optimization a filled function with only one parameter which satisfies this definition is presented we also discuss the properties of the proposed function and give a filled function method to solve the nonlinear integer programming problem the implementation of the algorithm on several test problems is reported with satisfactory numerical results

a formula for the kronecker products of schur functions of hook shapes abstract the kronecker product of two homogeneous symmetric polynomials p 1 and p 2 is defined by means of the frobenius map by the formula p 1 p 2 f f 1 p 1 f 1 p 2 when p 1 and p 2 are schur functions s i and s j respectively then the resulting product s i s j is the frobenius characteristic of the tensor product of the irreducible representations of the symmetric group corresponding to the diagrams i and j taking the scaler product of s i s j with a third schur function s k gives the so called kronecker coefficient c ijk s i s j s k which gives the multiplicity of the representation corresponding to k in the tensor product in this paper we give an explicit formula for c ijk where i and j are hooks and k is arbitrary

a new filled function for unconstrained global optimization abstract seeking global optima of an unconstrained and multi modal global optimization problem min x f x by constructing a filled function is concerned in this paper on the basis of analyzing filled functions presented before a new filled function is proposed and it is proved to meet the properties of a filled function moreover solutions of numerical experiments show that the function is quite effective

schur function analysis of the unitary discrete series representations of the non compact symplectic group the authors develop and analyse certain character reductions for the infinite dimensional unitary discrete series representations of the non compact symplectic group sp 2n r the group reductions considered are sp 2k r contains implies sp 2 r o k and the more general sp 2nk r contains implies sp 2n r o k they use schur function techniques to derive succinct formulae involving certain infinite series of schur functions the results are relevant to the study of many body systems with interactions of bilinear form and to the description of various quantum phenomena including collective behaviour

the complexity of finding minimum length generator sequences abstract the computational complexity of the following problem is investigated given a permutation group specified as a set of generators and a single target permutation which is a member of the group what is the shortest expression for the target permutation in terms of the generators the general problem is demonstrated to be psppace complete and indeed is shown to remain so even when the generator set is restricted to contain only two permutations the restriction on generator set cardinality is the best possible as the problem becomes soluble in polynomial time if the generator set cantains only one permutation an interesting feature of this problem is that it does not fall under the headings of two person games or formal languages which cover the great majority of known pspace complete problems some restricted versions of the problem in which the generator set is fixed rather than being part of the problem instance are also investigated and shown to be computationally tractable one result of this kind is that determining the most compact expression of a permutation in terms of cyclicly adjacent transpositions can be achieved in polynomial time thus from an initial arrangement of distinct objects on a circle one can quickly compute the smallest number of interchanges of adjacent objects required to realise any other arrangement surprisingly this problem appears substantially more difficult to solve than the related one for which a solution has been known for some time in which the objects are arranged on a line segment

a solution to the su n external state labelling problem based upon a u n 1 n 1 group i general theory the state labelling problem arising in the reduction of the direct product of a p positive row u n irreducible representation h1 hp0 with a q negative row one 0 h q h 1 into a sum of mixed u n irreducible representations k1 kp0 k q k 1 is solved by using the complementarity between u n and u p q within some positive discrete series irreducible representations of u pn qn this complementarity allows analysis of the problem in terms of the group chain u p q contains implies u p u q instead of u n u n contains implies u n for the most general su n irreducible representations corresponding to p q n 1 the relevant group chain is therefore u n 1 n 1 contains implies u n 1 u n 1 in such a case the additional labels include those of an intermediate u n 1 irreducible representation h1s hn 1s as well as the additional labels solving the state labelling problems for the products k1 kn 1 h1s hn 1s and k 1 k n 1 h1s hn 1s of u n 1 irreducible representations hence the proposed solution reflects in a direct way the operation of king s branching rule for the chain u n u n contains implies u n supplemented whenever necessary with king s modification rule

analogies between finite dimensional irreducible representations of so 2n and infinite dimensional irreducible representations of sp 2n r ii plethysms the basic spin difference character of so 2n is a useful device in dealing with characters of irreducible spinor representations of so 2n it is shown here that its kth fold symmetrized powers or plethysms associated with partitions of k factorize in such a way that r where r is the frobenius rank of the analogy between so 2n and sp 2n r is shown to be such that the plethysms of the basic harmonic or metaplectic character of sp 2n r factorize in the same way to give r moreover the analogy is shown to extend to the explicit decompositions into characters of irreducible representations of so 2n and sp 2n r not only for the plethysms themselves but also for their factors and explicit formulas are derived for each of these decompositions expressed in terms of various group subgroup branching rule multiplicities particularly those defined by the restriction from o k to the symmetric group sk illustrative examples are included as well as a

coherent states for the harmonic series irreducible representations of u p q i unitary operator coherent states unitary operator coherent states as defined by klauder 1963 perelomov 1972 and gilmore 1974 are considered for the harmonic series irreducible representations of u p q various properties of these states are reviewed including their reproducing kernel and measure explicit forms as well as the representation they lead to for the u p q generators starting from a contraction of the su p q lie algebra both dyson and holstein primakoff boson realisations of u p q are obtained in terms of pq pairs of boson creation annihilation operators and of the generators of a u p u q intrinsic group such boson realisations are applied to determine the matrix elements of the u p q generators between discrete bases classified according to the chain u p q contains implies u p u q finally the isomorphism between so 4 2 and su 2 2 is employed to derive the corresponding properties of the so 4 2 unitary operator coherent states for application in atomic physics

vertex operators and symmetric functions an algorithm for the calculation of q dependent spin characters of the symmetric group is given with an explicit example of s4 a method for constructing the q analogue of the vertex operators is developed a 1 1 correspondence between the space v of twisted q vertex operators and the ring of q deformed symmetric functions lambda x zq q t is established and a mapping from v to lambda x zq q t is defined a number of relevant theorems are given

a new filled function method for global optimization a new definition of the filled function is presented in this paper based on the definition a new filled function is presented and a global optimization algorithm is developed the implementation of the algorithm on several test problems is reported with satisfactory numerical results the numerical experiments further show that the algorithm may solve higher dimensional global optimization problems

vertex operators symmetric functions and the spin group γn abstract this work provides a vertex operator approach to the symmetric group sn and its double covering group n by generalizing a result of frenkel and sato for sn we formulate a correspondence between the space v of certain twisted vertex operators the ring of symmetric functions over q 2 and the space of nontrivial irreducible characters of n under this identification we show that a distinguished orthogonal basis of v corresponds to the set of nontrivial irreducible characters of n where both are parametrized by partitions with odd integer parts the counterpart of this distinguished basis in the ring over q 2 is the set of schur s q functions which are loosely speaking the square roots of the schur functions the nontrivial part of the character table of n is shown to be given by certain matrix coefficients in v

capacity of the trapdoor channel with feedback we establish that the feedback capacity of the trapdoor channel is the logarithm of the golden ratio and provide a simple communication scheme that achieves capacity as part of the analysis we formulate a class of dynamic programs that characterize capacities of unifilar finite state channels the trapdoor channel is an instance that admits a simple closed form solution

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

finding global minimizer with one parameter filled function on unconstrained global optimization abstract in this paper a filled function with one parameter is proposed for solving unconstrained global optimization problems this filled function is modified from the previously reported one see paper y j yang y l shang a new filled function method for unconstrained global optimization applied mathematics and computations 173 1 2006 501 512 the properties of the filled function are discussed in this paper an algorithm for unconstrained global optimization is developed from this new filled function the computational results show that this algorithm is efficient and reliable

the q deformation of symmetric functions and the symmetric group the q deformation of symmetric functions is introduced leading to q analogues of many well known relationships in the theory of symmetric functions q deformed scalar products are developed and used to define q dependent symmetric functions the symmetric functions commonly associated with the names hall littlewood schur and jack are all special cases of the q deformation of macdonald s 1988 new symmetric functions plambda s t a q analogue of the spin and ordinary characters of sn is given and illustrated by the explicit calculation of examples of q deformed characters the methods used are closely parallel to those of quantum groups

complementarity relation between the u p q and u n lie groups and some applications to atomic physics the complementarity relation between the unitary groups u d and u n within the symmetrical irreducible representations of the larger unitary group u dn is extended to non compact groups it is proved that the pseudo unitary group u p q is complementary with respect to u n within some positive discrete series irreducible representations of the larger pseudo unitary group u pn qn the latter arise when reducing the metaplectic irreducible representations 1 2 dn and 1 2 dn 13 2 of the real symplectic group sp 2dn r where d p q and they are characterised by a single label the eigenvalue of the first order casimir operator some applications of the u p q u n complementarity to atomic physics are outlined for such purposes the isomorphism between the lie algebras of su 2 2 and so 4 2 is used extensively the mathematical framework underlying the kibler negadi approach 1983 4 of the hydrogen atom dynamical group is extended to the independent electron dynamical group of intrashell many electron states as well as to the correlated electron dynamical group of intrashell doubly excited states

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

sorting of permutations by cost constrained transpositions the problem of finding a minimum decomposition of a permutation in terms of transpositions with predetermined non uniform and non negative costs is addressed alternatively computing the transposition distance between two permutations where transpositions are endowed with arbitrary non negative costs is studied for such cost functions polynomial time constant approximation decomposition algorithms are described for metric path costs exact polynomial time decomposition algorithms are presented the algorithms in this paper represent a combination of viterbi type algorithms and graph search techniques for minimizing the cost of individual transpositions and dynamic programing algorithms for finding minimum cost decompositions of cycles the presented algorithms have a myriad of applications in information theory bioinformatics and algebra

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

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

the symmetric group branching rules products and plethysms for spin representations it is shown that the symmetric group sn may be usefully embedded in the orthogonal group on and that this embedding leads directly to an n independent reduced notation for both the spin and ordinary representations of sn making use of this embedding together with the properties of q functions or hall littlewood functions branching rules for on spin down sn are developed and the general rule for the decomposition of spin representations under sn spin down sn 1 is obtained simple methods are given for calculating all possible kronecker products involving the spin and ordinary representations of sn and the resolution of kronecker squares into their symmetric and antisymmetric parts the spin representations of sn are systematically classified as to their orthogonal symplectic or complex characters the emphasis throughout is on obtaining results that obviate the need for explicit character tables and presenting results in an n independent manner as much as possible

a filled function method for finding a global minimizer of a function of several variables the concept of a filled function is introduced we construct a particular filled function and analyze its properties an algorithm for global minimization is generated based on the concept and properties of the filled function some typical examples with 1 to 10 variables are tested and computational results show that in most cases this algorithm works better than the tunneling algorithm the advantages and disadvantages are analyzed and further research directions are discussed

codes in permutations and error correction for rank modulation codes for rank modulation have been recently proposed as a means of protecting flash memory devices from errors we study basic coding theoretic problems for such codes representing them as subsets of the set of permutations of n elements equipped with the kendall tau distance we derive several lower and upper bounds on the size of codes these bounds enable us to establish the exact scaling of the size of optimal codes for large values of n we also show the existence of codes whose size is within a constant factor of the sphere packing bound for any fixed number of errors

a class of generalized filled functions with improved computability the filled function method ffm is an approach to find the global minimum of multimodal functions the computability of conventional filled functions is limited as they are defined on either exponential or logarithmic terms this paper proposes a new filled function that does not have such disadvantages numerical experiments on typical testing functions showed that the new function is superior to the conventional one

parisi s mean field solution for spin glasses as an analytic continuation in the replica number parisi s replica symmetry breaking solution for spin glasses is extended to finite replica number n the free energy fp n obtained this way as well as its first two derivatives with respect to n are shown to join the corresponding values in the sherrington kirkpatrick sk solution at a characteristic value ns t where stability breaks down in the latter the continuation composed of the sk branch fsk n for n or ns t and the parisi branch fp n for 0 or n or ns t fulfils the requirements of convexity monotonicity and stability for all n

the so 6 2 model of su 3 and its generalisation to su n the complementarity relation between the u p q and u n lie groups enables the authors to reformulate the so 6 2 model of su 3 in terms of a u 1 1 group and to outline its generalisation to a family of n 2 models of su n for n or 3 respectively associated with u n 2 1 u n 1 2 and u 1 n 2 groups

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

a solution to the su n external state labelling problem based upon a u n 1 n 1 group ii recursion relations for the su 2 and su 3 wigner coefficients for pt i see ibid vol 20 p 769 84 1987 the new solution to the su n external state labelling problem proposed in the first paper of this series is analysed in detail in the su 3 case in terms of the two complementary chains u 3 u 3 contains implies u 3 and u 2 2 contains implies u 2 u 2 the classification of su 3 coupled states is completed by the labels of an intermediate u 2 irreducible representation hs hs1hs2 directly related to king s branching rule for u 3 u 3 u 3 for pedagogical purposes the su 2 coupled state construction based upon the complementary chains u 2 u 2 contains implies u 2 and u 1 1 contains implies u 1 u 1 is considered first for both su 2 and su 3 it is proved that in addition to the standard recursion relations the corresponding wigner coefficients satisfy recursion relations of a different type arising from the action of the u 1 1 or u 2 2 complementary group generators a detailed example shows that such complementary recursion relations are quite useful for numerical purposes

critical behaviour in aperiodic systems the author considers layered systems with aperiodically modulated couplings and study the effect of inhomogeneity on the critical behaviour using scaling arguments a relevance irrelevance type criterion is formulated and for relevant inhomogeneities the scaling form of singular quantities is determined the author has performed exact calculations for the surface magnetization of aperiodic quantum ising chains and for a 1 1 dimensional directed polymer in a longitudinal aperiodic environment the results for both systems are in accord with scaling considerations for a marginal aperiodicity the critical exponents are found to be non universal

raising operators for the osp 1 2n r orthosymplectic lie superalgebras the branching rule is obtained for the decomposition of the osp 1 2n r oscillator like unitary irreducible representations into sp 2n r representations raising operators for osp 1 2n r sp 2n r are determined and used to construct a basis in the irreducible representation carrier space the matrix elements of the odd generators between two sp 2n r lowest weight states are calculated the general results so obtained are illustrated with the osp 1 4 r example

the remarkable algebra so 2n its representations its clifford algebra and potential applications properties of the real lie algebra so 2n and its finite dimensional representations are described the structure of the clifford algebras associated with the two fundamental spinor representations are determined and it is found in particular that for so 4n there is a remarkable asymmetry one representation is real the other complex though equivalent to its complex conjugate and the associated clifford algebras are inequivalent generalised dirac matrices are defined acting within the direct sum of these two spinor representations of so 2n these matrices generate a different clifford algebra not determined uniquely by those representations and their commutators do not span a representation of so 2n the isomorphisms of so 6 with su 3 1 and of so 8 with so 6 2 are described together with the corresponding mappings between finite dimensional irreducible representations a possible application to unified field theory is discussed

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

q functions and on to sn branching rules for ordinary and spin irreps following morris 1977 and luan dehuai and wybourne 1981 a simple method is given for the embedding on to sn of ordinary and spin irreps in both n dependent notation and in an n independent reduced notation basic spin irrep and ordinary irreps are combined using the properties of q functions and raising operators in order to give a complete set of branching rules of on to sn for spin irreps the modification rules for q functions given by morris are redefined to yield a complete and unambiguous set of rules

vertex operators and hall littlewood symmetric functions abstract we consider vertex operators on space v with a parameter t their components form an associative algebra which is a generalization of the clifford algebra a distinguished orthogonal basis of v is proved to be the hall littlewood symmetric functions we show that kostka foulkes polynomials or certain kazhdan lusztig polynomials for the affine weyl group of type a are matrix coefficients on the space v we also obtain certain generating functions for the product of hall littlewood functions and the kostka foulkes polynomials

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

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

finding discrete global minima with a filled function for integer programming the filled function method is an approach to find the global minimum of multidimensional functions this paper proposes a new definition of the filled function for integer programming problem a filled function which satisfies this definition is presented furthermore we discuss the properties of the filled function and design a new filled function algorithm numerical experiments on several test problems with up to 50 integer variables have demonstrated the applicability and efficiency of the proposed method

an axiomatic approach to constructing distances for rank comparison and aggregation we propose a new family of distance measures on rankings derived through an axiomatic approach that consider the nonuniform relevance of the top and bottom of ordered lists and similarities between candidates the proposed distance functions include specialized weighted versions of the kendall distance and the cayley distance and are suitable for comparing rankings in a number of applications including information retrieval and rank aggregation in addition to proposing the distance measures and providing the theoretical underpinnings for their applications we also analyze algorithmic and computational aspects of weighted distance based rank aggregation we present an aggregation method based on approximating weighted distance measures by a generalized version of spearman s footrule distance as well as a markov chain method inspired by pagerank where transition probabilities of the markov chain reflect the chosen weighted distances

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

the globally convexized filled functions for global optimization a new filled function the globally convexized filled function is proposed the value of this function tends to positive infinity as the norm of the variable vector tends to infinity and it has no stationary point in the region in which the objective function value is greater than or equal to the least value found so far but it does have a minimizer not only along the radius direction from the minimizer with the least objective function value found so far like filled function found earlier by ge in the region in which the objective function value is less than the least value found so far but with a prefixed minimizer how to adjust parameters in the filled function is considered such a filled function can be appropriately combined with stochastic stopping rules proposed by boender rinnooy kan and timmer the computational results show that this algorithm is quite efficient and reliable

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

products and symmetrized powers of irreducible representations of and their associates the calculation of kronecker products and plethysms of the infinite dimensional harmonic series unitary irreducible representations of the non compact group is considered the complementarity of and o k is used to define associate irreducible representations of this leads to simple relationships between kronecker products and plethysms of irreducible representations of and those of their corresponding associate irreducible representations in the process of proving the validity of these previously conjectured relationships several new identities are found for plethysms involving infinite series of schur functions in addition a general formula for plethysms of arbitrary irreducible representations of is derived and its implementation is illustrated with a detailed example a remarkable analogy is then observed between plethysms of the basic harmonic irreducible representations of and those of the basic spin irreducible representations of so 2n

the paraboson fock space and unitary irreducible representations of the lie superalgebra mathfrak osp 1 2n it is known that the defining relations of the orthosymplectic lie superalgebra mathfrak osp 1 2n are equivalent to the defining triple relations of n pairs of paraboson operators b pm i in particular with the usual star conditions this implies that the parabosons of order p correspond to a unitary irreducible infinite dimensional lowest weight representation v p of mathfrak osp 1 2n apart from the simple cases p 1 or n 1 these representations had never been constructed due to computational difficulties despite their importance in the present paper we give an explicit and elegant construction of these representations v p and we present explicit actions or matrix elements of the mathfrak osp 1 2n generators the orthogonal basis vectors of v p are written in terms of gelfand zetlin patterns where the subalgebra mathfrak u n of mathfrak osp 1 2n plays a crucial role our results also lead to character formulas for these infinite dimensional mathfrak osp 1 2n representations furthermore by considering the branching mathfrak osp 1 2n supset mathfrak sp 2n supset mathfrak u n we find explicit infinite dimensional unitary irreducible lowest weight representations of mathfrak sp 2n and their characters

analytic representations in quantum mechanics various euclidean hyperbolic and elliptic analytic representations are introduced and relations among them are discussed the bargmann analytic representation in the complex plane is considered and its relation to other phase space methods for the harmonic oscillator is reviewed the general theory that relates the growth of analytic functions with the density of their zeros is applied to bargmann functions and it leads to theorems on the completeness of sequences of glauber coherent states two hyperbolic analytic representations in the unit disc based on su 1 1 coherent states and also on phase states are introduced a third analytic representation in the complex plane based on barut girardello states is also considered and transformations which relate it to the other ones are studied in the case of systems with finite dimensional hilbert space an elliptic analytic representation in the extended complex plane and also another analytic representation based on theta functions are introduced the berezin formalism in the euclidean hyperbolic and elliptic cases is discussed contour analytic representations in these three cases are also presented

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

simple unified proofs of four duality theorems duality relationships between the irreps irreducible representations of pairs of distinct commuting groups g1 and g2 on hilbert spaces of interest have long played important roles in the atomic and nuclear shell models in addition to the well known schur weyl duality the most widely used duality relationships are the so called unitary unitary orthogonal symplectic i e noncompact symplectic symplectic symplectic compact symplectics and orthogonal orthogonal dualities proofs of these dualities exist in the literature but most of them are not readily accessible to physicists or give little insight into how they might be used in practice this paper presents unified proofs of the above mentioned dualities based on the explicit construction of states which are simultaneously of extreme weight for the actions of both g1 and g2 the proofs expressed in language familiar to physicists are simple systematic and provide useful insights

vector coherent state theory of the noncompact orthosymplectic superalgebras i general theory vector coherent states are defined for the positive discrete series irreducible representations of the non compact orthosymplectic superalgebras osp p 2n r where p 2m or 2m 1 an orthonormal bargmann berezin basis symmetry adapted to osp p 2n r so p sp 2n r so p u n is constructed and used to develop the k matrix theory for osp p 2n r a general method is provided for determining the conditions of existence of star representations and of grade star representations in the osp 2 2n r case and the branching rule for their decomposition into a direct sum of so p sp 2n r irreducible representations as a by product it also enables the matrix elements of the odd generators between basis states of lowest weight so p u n irreducible representations to be calculated in a straightforward way

products and plethysms for the fundamental harmonic series representations of u p q we give the decomposition of the kronecker products and the symmetrized kronecker squares of all the fundamental representations of the harmonic series of unitary irreducible representations of u p q the results for u 2 2 are relevant to two electron hydrogenic like atoms

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

symmetrized squares and cubes of the fundamental unirreps of we establish the complete decompositions of the symmetrized kronecker squares of the basic harmonic representations of conjectured by grudzinski and wybourne grudzinski k and wybourne b g 1996 j phys a math gen 29 6631 41 and give analogous complete results for the symmetrized cubes equivalence relationships are shown to exist between certain pairs of plethysms

products and symmetrized powers of irreducible representations of the calculation of branching rules tensor products and plethysms of the infinite dimensional harmonic series unitary irreducible representations of the non compact group is considered and the duality between and sp 2k exploited the branching rule for the restriction of an arbitrary harmonic series irreducible representation of to u n is derived and the decomposition is given explicitly for each of the infinite number of fundamental harmonic series irreducible representations of whose direct sum constitutes the metaplectic representation h of a concise expression for the decomposition of tensor products is derived and a complete analysis of the terms in both and is given a general formula for plethysms of arbitrary irreducible representations of is derived and its implementation illustrated both by means of a detailed generic example and by a complete determination of the symmetric and antisymmetric terms of finally relationships that arise from the embedding of the product groups and in the metaplectic group mp 4nk are discussed