branching rules for the supergroup su n m from those of su n m the decomposition of representations of supergroups into representations of subgroups is needed in practical applications in this paper we set up and exploit a fruitful one to one correspondence between the lie group branching su n m su n su m u 1 and the supergroup branchings su n m su n su m u 1 and su n1 n2 m1 m2 su n1 m1 su n2 m2 u 1 a simple and useful prescription is discovered for obtaining the su n m branching rules from those of su n m for any representation a large class of examples sufficient for many physical applications we can foresee are explicitly worked out and tabulated

the generalized atypical supertableaux of the orthosymplectic groups osp 2 2p the classification and the interpretation of the young supertableaux of the orthosymplectic group osp 2 2p are given a particular emphasis is made on the generalized atypical supertableaux associated to nonfully reducible atypical representations

application of generating function techniques to lie superalgebras the construction of generating functions for multiplicities of irreducible representations of lie superalgebras from generating functions for characters is examined and applied in order to obtain polynomial tensors and branching rules problems arising because of the existence of typical and atypical representations are discussed in detail the techniques are applied to osp 1 2 spl 1 2 osp 3 2 and osp 4 2

transformations of some induced osp 3 2 modules in an so 3 sp 2 basis the modules of the orthosymplectic lie superalgebra osp 3 2 induced from finite dimensional irreducible submodules of the stability subalgebra so 3 gl 1 are investigated the corresponding infinite dimensional irreducible or indecomposable modules the kac modules and the related typical and atypical modules are studied in detail every such module is decomposed into a direct sum of either indecomposable or irreducible modules of the even subalgebra so 3 sp 2 for each of these infinite dimensional or finite dimensional irreducible or indecomposable modules relations are written down giving the transformations of the basis under the action of the algebra generators

almost sure exponential behavior of a directed polymer in a fractional brownian environment this paper studies the asymptotic behavior of a one dimensional directed polymer in a random medium the latter is represented by a gaussian field bh on r r with fractional brownian behavior in time hurst parameter h and arbitrary function valued behavior in space the partition function of such a polymer is u t eb exp 0tbh dr br here b is a continuous time nearest neighbor random walk on z with fixed intensity 2 defined on a complete probability space pb independent of bh the spatial covariance structure of bh is assumed to be homogeneous and periodic with period 2 for h 12 we prove that the upper and lower almost sure limits lim supt t 2hlogu t and lim inft t 2hlogt logu t are non trivial in the sense that they are bounded respectively above and below by finite strictly positive constants thus as h passes through 12 the exponential behavior of u t changes abruptly this can be considered as a phase transition phenomenon novel tools used in this paper include sub gaussian concentration theory via the malliavin calculus detailed analyses of the long range memory of fractional brownian motion and an almost superadditivity property

sharp estimation of the almost sure lyapunov exponent for the anderson model in continuous space in this article we study the exponential behavior of the continuous stochastic anderson model i e the solution of the stochastic partial differential equation u t x 1 0 t x u s x ds 0 t w ds x u s x when the spatial parameter x is continuous specifically x r and w is a gaussian field on r r that is brownian in time but whose spatial distribution is widely unrestricted we give a partial existence result of the lyapunov exponent defined as lim t t 1 log u t x furthermore we find upper and lower bounds for lim sup t t 1 log u t x and lim inf t t 1 log u t x respectively as functions of the diffusion constant which depend on the regularity of w in x our bounds are sharper work for a wider range of regularity scales and are significantly easier to prove than all previously known results when the uniform modulus of continuity of the process w is in the logarithmic scale our bounds are optimal

superdiffusivity for a brownian polymer in a continuous gaussian environment this paper provides information about the asymptotic behavior of a one dimensional brownian polymer in random medium represented by a gaussian field w on r r assumed to be white noise in time and function valued in space according to the behavior of the spatial covariance of w we give a lower bound on the power growth wandering exponent of the polymer when the time parameter goes to infinity the polymer is proved to be superdiffusive with a wandering exponent exceeding any 3 5

lyapunov exponent for the parabolic anderson model in rd abstract we consider the asymptotic almost sure behavior of the solution of the equation u t x u 0 x 2 0 t u s x d s 0 t u s x w x s where w x x r d is a field of brownian motions in fact we establish existence of the lyapunov exponent lim t 1 t log u t x we also show that c 1 1 3 c 2 1 5 as 0 under the assumption that the correlation function of the background field w x x r d is c for 1 2

character expansion for u n groups and u n m supergroups a formula is presented to generate expansions over u n characters the procedure to obtain the corresponding u n m character expansions is also described several examples are worked out in both cases

branching rules for a class of typical and atypical representations of gl m n the irreducible representations of the lie superalgebra gl m n with highest weights of the form 1 2 m are investigated using a recently introduced induced module construction for atypical modules the gl m n gl m n 1 branching rules are obtained and a suitable gel fand tsetlin basis is introduced the class of representations considered includes some multiply atypical irreducible representations of gl m n and all irreducible representations of gl m 1

a conjectured character formula for atypical irreducible modules of the lie superalgebra sl m n a formula is used for the character of each finite dimensional irreducible representation of the lie superalgebra sl m n this new formula appears to cover not only typical but also both singly and multiply atypical representations a comparison is made with previous conjectures and the problem of generalising the formula to cover all of the basic classical lie superalgebras is discussed

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

character formulas for irreducible modules of the lie superalgebras sl m n kac distinguished between typical and atypical finite dimensional irreducible representations of the lie superalgebras sl m n and provided an explicit character formula appropriate to all the typical representations here the range of validity of some character formulas for atypical representations that have been proposed are discussed several of them are of the kac weyl type but then it is proved that all formulas of this type fail to correctly give the character of one particular atypical representation of sl 3 4 having ruled out therefore all such formulas a completely new extension of the kac weyl character formula is proposed the validity of this formula in the case of all covariant tensor irreducible representations is proved and some evidence in support of the conjecture that it covers all irreducible representations of sl m n is presented

representations of osp m 2n and young supertableaux young supertableaux are introduced for finite dimensional representations of orthosymplectic superalgebras and used to study their content

branching rules for typical and atypical representations of gl n 1 the gl n 1 down arrow gl n gl 1 branching rules are determined for all finite dimensional irreducible typical and atypical representations of gl n mod 1 using a recently introduced induced module construction for atypical modules and confirm those found recently by palev 1988 by another method the lowest weights and characters of the irreducible representations are found and the validity for gl n mod 1 of the character formula conjectured by hughes and king 1987 is confirmed some generalisations and extensions of the method are discussed

skorohod integration and stochastic calculus beyond the fractional brownian scale abstract we extend the skorohod integral allowing integration with respect to gaussian processes that can be more irregular than any fractional brownian motion this is done by restricting the class of test random variables used to define skorohod integrability a detailed analysis of the size of this class is given it is proved to be non empty even for gaussian processes which are not continuous on any closed interval despite the extreme irregularity of these stochastic integrators the skorohod integral is shown to be uniquely defined and to be useful an ito formula is established it is employed to derive a tanaka formula for a corresponding local time linear additive and multiplicative stochastic differential equations are solved an analysis of existence for the stochastic heat equation is given

mixed supertableaux of the superunitary groups ii su n m the young supertableaux of the superunitary groups are studied we give the conditions on the size of legal supertableaux the type of atypicity for tensor representations which are irreducible making the link between the kac dynkin parameters and those of the supertableau a discussion is given on nonfully reducible representations and generalized atypical supertableaux

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

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

representations of supergroups an explicit construction of representations of supergroups is given in terms of direct products of covariant and contravariant fundamental representations the rules of supersymmetrization are characterized by extended young supertableaux this constructive approach leads to explicit transformation properties of higher representations as well as to closed explicit formulas for characters from which other invariants such as dimensions and eigenvalues of all casimir operators can be calculated we have applied this approach so far to the supergroups su n m osp n 2m p n for which we have obtained all the representations constructible as direct products of the fundamental defining representations an argument is presented toward the irreducibility of all these representations

anomalies and eigenvalues of casimir operators for lie groups and supergroups an expression for the anomaly of any representation of the su n groups and su n m supergroups is presented some anomaly free complex representations of su n and su n m are pointed out for su n these occur for n 5 and for large dimensions for su n m they already occur for low dimensions also a generating function to obtain the eigenvalues of all casimir operators of su n m is given and an extension to other supergroups is pointed out

coefficients of fractional parentage for u m p n q u m n u p q and u m n u m u n the outer product reduction coefficients orc which reduce the representation rep induced from the irreps of the permutation groups s f1 and s f2 into the irreps of s f1 f2 are shown to be the indirect coupling coefficients for the u m p n q u m n u p q irreducible basis the non standard orc for reducing the rep induced from the non standard irreps of s f13 s f1 s f3 and s f24 s f2 s f4 into that of s f s f12 s f34 with fij fi fj f f12 f34 are identified with the u f u f12 u f34 clebsch gordan coefficient for the special gel fand bases of u f12 and u f34 the u m p n q u m n u p q cfp as well as its special case the u m p u m u p cfp are identified with the s f s f12 s f34 outer product isoscalar factor the u m n u m u n cfp are obtained from the u m n u m u n cfp by simply changing all the partition labels for u n into their conjugates interchanging rows with columns and taking into account a phase change the cfp can be calculated from the orc numerical values of the one body cfp for systems with up to six particles are tabulated

finite dimensional representations of the lie superalgebra sl 1 3 in a gel fand zetlin basis ii nontypical representations in a series of two papers all finite dimensional irreducible representations of the special linear lie superalgebra sl 1 3 are constructed explicit formulas for the generators in an orthonormal gel fand zetlin basis of the even subalgebra gl 3 are given this paper develops a background for constructing the representations expressions for the transformation properties of the basis under the action of the generators are written down within all typical sl 1 3 modules

on the composition factors of kac modules for the lie superalgebras sl m n in the classification of finite dimensional modules of lie superalgebras kac distinguished between typical and atypical modules kac introduced an induced module the so called kac module v with highest weight which was shown to be simple if is a typical highest weight if is an atypical highest weight the kac module is indecomposable and the simple module v can be identified with a quotient module of v in the present paper the problem of determining the composition factors of the kac modules for the lie superalgebra sl m n is considered an algorithm is given to determine all these composition factors and conversely an algorithm is given to determine all the kac modules containing a given simple module as a composition factor the two algorithms are presented in the form of conjectures and illustrated by means of detailed examples strong evidence in support of the conjectures is provided the combinatorial way in which the two algorithms are intertwined is both surprising and interesting and is a convincing argument in favor of the solution to the composition factor problem presented here

representations of orthosymplectic superalgebras ii young diagrams and weight space techniques for pt i see ibid vol 16 p 473 1983 finite dimensional graded tensor representations of osp m n are enumerated via standard young diagrams and their corresponding highest weight and kac dynkin labels are given a uniform set of conditions on the diagram shape necessary and sufficient for atypicality are presented weight space techniques are used to provide a complete analysis of an atypical representations for the low rank cases osp 2 2 osp 3 2 and osp 4 2

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

on the long time behavior of the stochastic heat equation we consider the stochastic heat equation in one space dimension and compute for a particular choice of the initial datum the exact long time asymptotic in the carmona molchanov approach to intermittence in non stationary random media this corresponds to the identification of the sample lyapunov exponent equivalently by interpreting the solution as the partition function of a directed polymer in a random environment we obtain a weak law of large numbers for the quenched free energy the result agrees with the one obtained in the physical literature via the replica method the proof is based on a representation of the solution in terms of the weakly asymmetric exclusion process

structure of atypical representations of the lie superalgebras sl m n in contrast to lie algebras lie superalgebras contain typical and atypical irreducible representations character formulae for typical representations of classical lie superalgebras have been known for a long time but the structure of atypical representations remained unsolved the author constructs the character formula for atypical representations of the special linear lie superalgebras sl m n

unitary lowest weight representations of the noncompact supergroup osp 2n 2m r the oscillator construction of the unitary irreducible lowest highest weight representations of the noncompact supergroup osp 2n 2m r with even subgroup so 2n sp 2m r is given in particular simple rules for determining the so 2n sp 2m r decomposition of the unitary lowest weight representations of osp 2n 2m r are derived

on the stochastic heat equation with spatially colored random forcing we consider the stochastic heat equation of the following form t ut x lut x b ut x ut x t x for t 0 x r where l is the generator of a levy process and is a spatially colored temporally white gaussian noise we will be concerned mainly with the long term behavior of the mild solution to this stochastic pde for the most part we work under the assumptions that the initial data u0 is a bounded and measurable function and is nonconstant and lipschitz continuous in this case we find conditions under which the preceding stochastic pde admits a unique solution which is also weakly intermittent in addition we study the same equation in the case that lu is replaced by its massive dispersive analogue lu u where r and we describe accurately the effect of the parameter on the intermittence of the solution in the case that u is proportional to u the parabolic anderson model we also look at the linearized version of our stochastic pde that is the case when is identically equal to one any other constant works also in this case we study not only the existence and uniqueness of a solution but also the regularity of the solution when it exists and is unique

finite dimensional representations of the lie superalgebra gl 2 2 in a gl 2 gl 2 basis ii nontypical representations all finite dimensional irreducible representations of the general linear lie superalgebra gl 2 2 are written down in matrix form the basis within each representation space is chosen in such a way that it makes evident the decomposition of gl 2 2 into irreducible representations of its even subalgebra gl 2 gl 2 special attention is devoted to the analysis of all nontypical representations and some indecomposible representations

some results on complete lie superalgebras abstract a lie superalgebra g is called a complete if c g 0 and der g ad g this paper provides sufficient conditions for a lie superalgebra to be complete and uses this result to construct families of such algebras the basic results are a generalization of d j meng commun algebra 22 1994 5509

superunitary reduction of the orthosymplectic supertableaux the supertableaux of the orthosymplectic groups osp 2 2p and osp 2 1 2p are decomposed in sums of supertableaux of the superunitary group su p

quenched asymptotics for brownian motion of renormalized poisson potential and for the related parabolic anderson models let bs be a d dimensional brownian motion and dx be an independent poisson field on rd the almost sure asymptotics for the logarithmic moment generating function loge0exp t0v bs ds t are investigated in connection with the renormalized poisson potential of the form v x rd1 y x p dy dy x rd the investigation is motivated by some practical problems arising from the models of brownian motion in random media and from the parabolic anderson models

character expansions for the orthogonal and symplectic groups formulas for the expansion of arbitrary invariant group functions in terms of the characters for the sp 2n so 2n 1 and so 2n groups are derived using a combinatorial method the method is similar to one used by balantekin to expand group functions over the characters of the u n group all three expansions have been checked for all n by using them to calculate the known expansions of the generating function of the homogeneous symmetric functions an expansion of the exponential of the traces of group elements appearing in the finite volume gauge field partition functions is worked out for the orthogonal and symplectic groups

irreducible representations of the superalgebras type ii the representation of the orthosymplectic algebras and the other members of their class are built explicitly with simple techniques

atypical representations for type i lie superalgebras a new modified induced module construction is presented for all finite dimensional irreducible typical and atypical modules of a type i basic classical lie superalgebra the method is illustrated with some low dimensional representations of gl m mod n and all representations of gl 2 mod 1

on the brownian directed polymer in a gaussian random environment in this paper we introduce a model of brownian polymer in a continuous random environment the asymptotic behavior of the partition function associated to this polymer measure is studied and we are able to separate a weak and strong disorder regime under some reasonable assumptions on the spatial covariance of the environment some further developments concerning some concentration inequalities for the partition function are given for the weak disorder regime

irreducible representations of the exceptional lie superalgebras d 2 1 α the shift operator technique is used to give a complete analysis of all finite and infinite dimensional irreducible representations of the exceptional lie superalgebrasd 2 1 for all cases the star or grade star conditions for the algebra are investigated among the finite dimensional representations there are no star and only a few grade star representations but an infinite class of infinite dimensional star representations is found explicit expressions are given for the doublet representation of d 2 1 the one missing label problem d 2 1 su 2 su 2 su 2 is discussed in detail and solved explicitly

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

sharp gaussian regularity on the circle and applications to the fractional stochastic heat equation abstract a sharp regularity theory is established for homogeneous gaussian fields on the unit circle two types of characterizations for such a field to have a given almost sure uniform modulus of continuity are established in a general setting the first characterization relates the modulus to the field s canonical metric the full force of fernique s zero one laws and talagrand s theory of majorizing measures is required the second characterization ties the modulus to the field s random fourier series representation as an application it is shown that the fractional stochastic heat equation has up to a non random constant a given spatial modulus of continuity if and only if the same property holds for a fractional antiderivative of the equation s additive noise a random fourier series characterization is also given

on unitary lie superalgebras from the spin orbit supersymmetrisation procedure from the basis elements of a clifford algebra cl n the authors generate a grading leading to a unitary lie superalgebra when n is even such a construction is motivated by the understanding of the specific properties of the fermionic variables in the so called spin orbit coupling procedure of supersymmetrisation in n 2 supersymmetric quantum mechanics the n odd case is also considered and some specific examples are discussed

representations of the superalgebra f4 and young supertableaux the authors present a method for constructing typical as well as atypical finite dimensional representations of the superalgebra f4 for such a purpose young supertableaux are introduced

remark on the branching theorem and supersymmetric algebras supersymmetric algebras have already proved useful in giving trans parent proofs of a number of basic results of representation theory 3 7 9 1 l l 51 specifically the technique of introducing virtual variables which may have a different signature than the signature of variables to be delt with often cuts down the amount of computation furthermore the extension of results of representation theory to the superalgebraic setting sheds new light and permits us to establish natural correspondences that were formally missing in this note we carry out this program by deriving a superalgebraic version of the branching rules for the representations of the general linear group while the statement of supersymmetric branching rules is in all respects similar to the ordinary one and differing from it in our allowing variables of two signature the proof yields a useful dividend namely a simple combinatorial construction of a canonical basis for the decomposi tion of a restriction of a representation as an application we give a supersymmetric generalization of pieri s formula as well as a proof of this formula which is perhaps as short as it can be whittled down to this application has been inspired by some recent work of bofli s 61 we have benefited from the pioneering work of berele and regev 3 who were first to state such a supersymmetric extension of branching rules as well as from the insights of balentekin and bars 2 255

almost sure asymptotics for the continuous parabolic anderson model we consider the parabolic anderson problem t u u x u on d with initial condition u 0 x 1 here 0 is a diffusion constant and is a random homogeneous potential we concentrate on the two important cases of a gaussian potential and a shot noise poisson potential under some mild regularity assumptions we derive the second order term of the almost sure asymptotics of u t 0 as t