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

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

fast recursion formula for weight multiplicities the purpose of this note is to describe and prove a fast recursion formula for computing multiplicities of weights of finite dimensional representations of simple lie algebras over c until now information about weight multiplicities for all but some special cases 1 2 has had to be found from the recursion formulas of freudenthal 3 or racah 4 typically these formulas become too laborious to use for hand computations for ranks 5 and dimensions 1 0 0 and for ranks 10 and dimensions 10 on a large computer 5 6 with the proposed method the multiplicities can routinely be calculated even by hand for dimensions far exceeding these as an example we present a summary of calculations 7 of all multiplicities in the first sixteen irreducible representations of es let be a semisimple lie algebra over c with root system a and weyl group w relative to a cartan subalgebra let a be the positive roots with respect to some ordering and ii at a the set of simple roots let q and p be the root and weight lattices respectively spanning the real vector space f c if x c p we denote by x the set of dominant elements of x relative to n let m be an irreducible module with highest weight a and weight system 2 an important feature of the approach is the direct determination of 2 without computing outside the dominant chamber since every w orbit is represented by one weight x 2 of the same multiplicity it suffices to compute such x s the recursion formula for computing the multiplicities is a modification proposition 4 of the freudenthal formula in which the weyl group has been exploited to collapse it as much as possible after describing the procedure we present the e example finally the necessary proofs are given

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

monoids and direct sum decompositions over local rings abstract let r be a local ring order i e a one dimensional local noetherian ring whose completion r is reduced and let m be a finitely generated r module we consider two monoids m which consists of the isomorphism classes of r modules which arise as direct summands of direct sums of finitely many copies of m and m which consists of the n tuples b 1 b n such that the r module i 1 n b i v i is extended from an r module where v 1 v n are the distinct and uniquely determined indecomposable direct summands of the r module m here bv denotes the direct sum of b copies of v and n 0 1 2 the monoids m and m are isomorphic and we show that m ker a n n for some integer matrix a z d n monoids which are isomorphic to ker a n n for some a z d n are called positive normal in r wiegand j algebra in press it is shown that given a positive normal monoid there exist a local ring order domain r and finitely generated torsion free r module m such that m we show that given a local ring order r there exists a positive normal monoid such that for each finitely generated r module m m is not isomorphic to the proof depends on the fact that there exist rank three positive normal monoids with arbitrarily large embedding dimension where the embedding dimension of is defined as the smallest n such that ker a n n where a z d n

critical behaviour of the random field ising model we present an analytic real space renormalization group calculation for the random field ising model we apply the migdal kadanoff approximation for the renormalization of a cubic cell in dimensions d introducing a new field partitioning scheme which allows us to treat the random field fluctuations in a coherent manner our scheme leads naturally to a lower critical dimensionality and allows us to calculate a complete set of three independent exponents in arbitrary dimension in three dimensions the magnetization exponent and the schwartz soffer inequality is almost satisfied as an equality we expand analytically in further we show that and the magnitude of the inequality go to zero exponentially with we calculate the crossover exponent from pure to the random field system and find surprisingly good agreement with experimental values we find that satisfies the schwartz soffer inequality the susceptibility exponent of the pure system we expand in and find that the magnitude of the inequality varies exponentially in finally we find that dimensional reduction is satisfied to first order in with the reduced dimension

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

direct sum decompositions over local rings let r m be a local ring commutative and noetherian if r is complete or more generally henselian one has the krull schmidt uniqueness theorem for direct sums of indecomposable finitely generated r modules by passing to the m adic completion r we can get a measure of how badly the krull schmidt theorem can fail for a more general local ring we assign to each finitely generated r module m a full submonoid m of nn where n is the number of distinct indecomposable direct summands of r rm this monoid is naturally isomorphic to the monoid m of isomorphism classes of modules that are direct summands of direct sums of finitely many copies of m the main theorem of this paper states that every full submonoid of nn arises in this fashion moreover the local ring r realizing a given full submonoid can always be taken to be a two dimensional unique factorization domain the theorem has two non commutative consequences 1 a new proof of a recent theorem of facchini and herbera characterizing the monoid of isomorphism classes of finitely generated projective right modules over a non commutative semilocal ring and 2 a characterization of the monoids n where n is an artinian right module over an arbitrary ring

a finite subgroup of the exceptional lie group g2 with a view to further refining the use of the exceptional group g2 in atomic and nuclear spectroscopy it is confirmed that a simple finite subgroup l168 psl2 7 of order 168 of the symmetric group s8 is also a subgroup of g2 it is established by character theoretic and other methods that there are two distinct embeddings of l168 in g2 analogous to the two distinct embeddings of so 3 in g2 relevant branching rules tensor products and symmetrized tensor products are tabulated as a stimulus to further applications the branching rules are given for the restriction from l168 to the octahedral crystallographic point group o

the place of the adjoint representation in the kronecker square of irreducible representations of simple lie groups the multiplicity of occurrence of the adjoint representation in the decomposition of the square of any finite dimensional irreducible representation of any compact simple lie group is shown to be equal to the number of non vanishing components of the dynkin label of the resolution of this multiplicity into contributions to the symmetric and antisymmetric squares of is discussed with complete results being found for all of the classical and some of the exceptional simple lie groups and partial results culminating in conjectures for the remaining exceptional groups

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

generating functions and elementary young tableaux when constructing polynomial bases for group representations one is led naturally to consider a stretched product of states elementary multiplets are states that cannot be expressed as the stretched product of any other two states and they can be identified with terms occurring in certain generating functions in this paper the authors define an analogous stretched product of young tableaux together with elementary tableaux these may be used in a manner similar to elementary multiplets to construct generating functions they consider several examples where this method can be applied and in particular give new large n rank free branching rule generating functions for the group subgroup pairs su n su m contained in implied by su n m first four labels non zero so n contained in implied by su n first three labels non zero and sp 2n contained in implied by su 2n first five labels non zero this method of finding these generating functions uses the theory of grobner bases which allows the authors to make precise the notion of compatibility relations or forbidden products of elementary tableaux

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

signatures of finite exceptional lie algebra representations the paper deals with the real exceptional lie algebras of types ei i 6 7 8 and their arbitrary irreducible representations hermitian forms which are invariant relative to these representations are considered signature formulas for these forms are obtained

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

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

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

standard young tableaux and weight multiplicities of the classical lie groups by examining the branching rules for all irreducible representations of the classical groups u k su k so 2k 1 sp 2k and so 2k on restriction to u 1 u 1 u 1 standard young tableaux are specified for each of these groups it is shown that these tableaux determine the corresponding characters of the irreducible representations the rules for constructing these tableaux are derived and in this way the determination of weight multiplicities is reduced to a simple combinatorial exercise general formula for such weight multiplicities are given encompassing the most difficult case namely that of so 2k illustrative examples are provided including some yielding the explicit k dependence of weight multiplicities

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

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

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 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

signatures of finite eiii evi evii eix representations the paper deals with the real exceptional lie algebras eiii evi evii eix and their arbitrary irreducible representations hermitian forms invariant relative to these representations are considered the existing signature formulae for these forms are improved

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

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

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

orbit depths of affine kac moody algebras since the number of weights in a highest weight representation of an affine algebra is infinite it is particularly useful for these algebras to classify weight systems in terms of weyl orbits it is shown that even though depths in affine orbits are unbounded any weight depth may be determined in a number of steps that is the same for all weights the concepts of root layers and families and partner roots are introduced and used in the procedure a projection of weight vectors onto a euclidean subspace is used to present a simple geometrical picture of orbits

absolute minima of the higgs potential for the 75 of su 5 the problem of minimising the higgs potential for the 75 dimensional representation of su 5 is considered the most general form of the potential is established in the absence of a cubic term it is shown that the possible symmetry breaking patterns include su 3 su 2 u 1 and a new counterexample to michel s conjecture 1979 is found

modification rules for orthosymplectic superalgebras modification rules are presented for finite dimensional graded tensor representations of osp m n with the representations specified by young supertableaux these rules relate non standard supertableaux to standard supertableaux via the removal of continuous boundary hooks all typical tensor representations are treated together with atypical representations which satisfy up to two atypically conditions

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

multiplicity free tensor products of irreducible representations of the exceptional lie groups for each of the exceptional lie groups a complete determination is given of those pairs of finite dimensional irreducible representations whose tensor products or squares may be resolved into irreducible representations that are multiplicity free i e such that no irreducible representation occurs in the decomposition of the tensor product more than once explicit formulae are presented for the decomposition of all those tensor products that are multiplicity free many of which exhibit a stability property

real space renormalization group analysis of the random field ising model we present an analytic migdal kadanoff renormalization group analysis of the random field ising model the renormalization flows to a zero temperature critical point from which we calculate independently three critical exponents in arbitrary dimension in three dimensions the magnetization exponent and the schwartz soffer inequality is almost satisfied as an equality expanding analytically in we find that and the distance from the upper bound of the equality go to zero exponentially with

branching rules for irreducible representations of e8 into d8 the branching rules for irreducible representations of e8 into d8 are calculated using the knowledge of the kronecker products for those two algebras tables of kronecker products for both e8 and d8 algebras are also included

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

on descent in dimension two and non split gorenstein modules in this paper we present a ring with a gorenstein module but no dualizing module the context of this example will be illustrated in the historical overview presented below in 21 sharp furnished the definition of a gorenstein module g of rank t over a local finite dimensional ring a g is finitely generated and for p spec a the bass number ll z g equals t if i hta fi and equals zero otherwise sharp went further to show that any ring admitting a gorenstein module must be cohen macaulay and also must have gorenstein formal fibres 21 we observe that a gorenstein module g of rank one over an n dimensional local ring a m k induces a natural equivalence of functors ext g hom e k on modules of finite length for this reason fossum foxby griffith and reiten assigned to gorenstein modules of rank one the name dualizing module s in the same paper it was obtained that any gorenstein a module is a direct sum of copies of the gorenstein a module of minimal rank of which up to isomorphism there is only one s it is clear that if a ring has a dualizing module then any of its gorenstein modules are direct sums of the dualizing module foxby s reiten 19 and sharp 21 showed that a local cohenmacaulay ring a admits a dualizing complex if and only if a has a dualizing module on the other hand grothendieck who initiated the notion of a dualizing complex provided the following result

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

symmetrized powers of the fundamental irrep of e6 the resolution of all possible antisymmetrically symmetrized powers of the fundamental irrep 27 is given this reduces the problem of plethysms of the fundamental irrep to the evaluation of ordinary kronecker products of e6 irreps

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

some noteworthy spin plethysms the spin plethysms lambda g x delta that arise in the reduction of delta under so n to g when 1 to lambda g are considered it is shown that for the simple lie algebras of rank k if lambda g phi g the adjoint representation of g then phi g x delta 2 k2 delta g where delta g is the representation of g whose highest weight is half the sum of the positive roots certain results for other representations are described a remarkable series of s functions is introduced leading to a new dimensional equality between certain representations of o 2k and sp 2k

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

lack of self averaging in random systems liability or asset abstract the study of quenched random systems is facilitated by the idea that the ensemble averages describe the thermal averages for any specific realization of the couplings provided that the system is large enough careful examination suggests that this idea might have a flaw when the correlation length becomes of the order of the size of the system we find that certain bounded quantities are not self averaging when the correlation length becomes of the order of the size of the system this suggests that the lack of self averaging expressed in terms of properly chosen signal to noise ratios may serve to identify phase boundaries this is demonstrated by using such signal to noise ratios to identify the boundary of the ferromagnetic phase of the random field ising system and compare the findings with more traditional measures

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

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

young supertableaux of the basic lie superalgebras among the simple lie superalgebras lsa the basic classical ones are well known together with many of their finite dimensional representations this is particularly true for the typical representations which are known to be completely reducible it is well known that reducible but indecomposable representations of the basic lsa are not completely characterised by their highest weight hw or the eigenvalues of their casimir operators in fact there may be several different finite representations characterised by the same hw such representations are called non typical or atypical the aim of this paper is to establish a complete characterisation of the typical and non typical representations by means of properly defined young supertableaux yst therefore the definition we are going to use for our yst differs slightly from the standard definitions the examples given throughout this paper help the illustration of this new interpretation for the yst which makes possible the complete classification of the indecom posable representations characterised by a generalised notion of hw

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

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

geometrical classification scheme for weights of lie algebras every weight of a lie algebra belongs to one particular weyl orbit the depth within an orbit has been defined as the minimum number of weyl reflections associated with simple roots that are necessary to transform the weight into the dominant weight of the orbit a simple set of classification parameters that measures the depth and distinguishes between different weights of the same depth is introduced the parameters are geometrical in that they are related directly to the orientation of the weight vector they correspond to a particular member of the weyl group that transforms the weight into the dominant weight a simple rule is given for listing all the weights of an orbit in terms of the parameters

rank dependency of weight multiplicities for simple lie groups the determination of the rank dependence of the weight multiplicities for the groups o2k 1 sp2k and o2k is simplified using the properties of schur functions together with a simple weight space mapping the weight multiplicities are factorised into products of much simpler quantities whose rank dependence is readily computed the spinor weight multiplicities for so2k are shown to be simply related to those of o2k the weight multiplicities for the k part tensor representation of so2k are examined in terms of difference characters some specific results are tabulated for so2k

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

the weyl orbits of g2 f4 e6 and e7 simple methods are described for constructing the weyl orbits of exceptional lie algebras and for identifying the orbit of an arbitrary weight the methods make use of convenient bases in weight space they are applied to g2 f4 e6 and e7 a complete table of all weights of all orbits of f4 is given the weights of the 37 orbits of smallest weight length of e6 and e7 are given in tabular form the depth structures of the orbits are discussed

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

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

real space renormalization group approach to the random field ising model the authors discuss results from two types of real space renormalization group rsrg calculations applied to the random field ising model in three dimensions starting from a lattice of size l the rsrg is used to reduce the lattice to a size l 2 on which the trace is done exactly in this way thermodynamic properties such as the magnetization and susceptibility can be determined approximately they find that for a given size the susceptibility increases as the temperature t is reduced down to the transition temperature tc and becomes essentially independent of temperature below tc both in the vicinity of tc and at lower temperatures there are large sample to sample fluctuations in the susceptibility which grow with increasing system size they interpret these results in terms of the droplet theory of the transition

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