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

explicit expressions of the nth gradient of 1 r explicit expressions of the nth gradient tensor and partial derivatives of the function 1 r are given in the absolute and in the local bases associated respectively with the cartesian and the spherical coordinates of space they may be used in various multipolar developments of r 1 interaction energies

classification of finite dimensional modules of singly atypical type over the lie superalgebras sl m n we classify the finite dimensional indecomposable sl m n modules with at least a typical or singly atypical primitive weight we do this classification not only for weight modules but also for generalized weight modules we obtain that such a generalized weight module is simply a module obtained by linking subquotient modules of generalized kac modules by applying our results to sl m 1 we have in fact completely classified all finite dimensional sl m 1 modules

highest weight representations for gl m n and gl m n it is shown how vector coherent state vcs theory enlightens the simultaneous discussion of representation theory for classical lie algebras and superalgebras and provides an optimal framework for the explanation of the noted similarities and dissimilarities of their representations reducibility atypicality and the positive definitiveness of inner products in vcs representation spaces are discussed the discussion is exemplified through the parallel and explicit construction of highest weight ladder representations of the lie algebra gl m n and superalgebra gl m n in gl m gl n bases

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

character and dimension formulae for general linear superalgebra the generalized kazhdan lusztig polynomials for the finite dimensional irreducible representations of the general linear superalgebra are computed explicitly the result is then applied to prove a conjectural character formula put forward by van der jeugt et al in the late 80s we simplify this character formula to cast it into the kac weyl form and derive from it a closed formula for the dimension of any finite dimensional irreducible representation of the general linear superalgebra

finite dimensional representations of the quantum superalgebra uq gl 3 2 in a reduced uq gl 3 2 contains implies uq gl 3 1 contains implies uq gl 3 basis for generic q we give expressions for the transformations of all essentially typical finite dimensional modules of the hopf superalgebra uq ql 3 2 the latter is a deformation of the universal enveloping algebra of the lie superalgebra gl 3 2 the basis within each module is similar to the gel fand zetlin basis for gl 5 we write down expressions for the transformations of the basis under the action of the chevalley generators

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

finite dimensional irreducible representations of the lie superalgebra sl 1 3 in a gel fand zetlin basis a concept of a gel fand zetlin pattern for the lie superalgebra sl 1 3 is introduced within every finite dimensional irreducible sl 1 3 module the set of the gel fand zetlin patterns constitute an orthonormed basis called a gel fand zetlin basis expressions for the transformation of this basis under the action of the generators are written down for every finite dimensional irreducible representation

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

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

primitive vectors of kac modules of the lie superalgebras sl m n it was conjectured by hughes et al j math phys 33 470 491 1992 that there exists a bijection between the composition factors of a kac module and the so called permissible codes in a previous paper it was proved that to any unlinked code there corresponds a composition factor of the kac module here it is proved that to any linked code there corresponds a composition factor of the kac module

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

some simple representations of the affine lie superalgebra a 1 1 0 in this paper some simple modules of the affine lie superalgebra a 1 1 0 are considered which although they are neither highest nor lowest weight modules are constructed in a manner closely analogous to the construction of highest weight modules of the finite dimensional lie superalgebras a m n

highest weight irreducible representations of the lie superalgebra gl 1 two classes of irreducible highest weight modules of the general linear lie superalgebra gl 1 are constructed within each module a basis is introduced and the transformation relations of the basis under the action of the algebra generators are written down

character and dimension formulae for queer lie superalgebra closed formulae are constructed for the characters and dimensions of the finite dimensional simple modules of the queer lie superalgebra mathfrak q n this is achieved by refining brundan s algorithm for computing simple mathfrak q n characters

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

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

distributional analysis for discontinuous fields abstract the jump conditions that hold across singular surfaces for the fields having step function discontinuities do not in general apply if these surfaces themselves carry concentrated fields in this note the general situation when the surfaces of discontinuity carry multilayers and deform as they propagate is discussed formulas are presented for the first and second derivatives for these multilayers

irreducible finite dimensional representations of the lie superalgebra gl n 1 in a gel fand zetlin basis all finite dimensional irreducible representations of the general linear lie superalgebra gl n 1 are studied for each representation a concept of a gel fand zetlin basis is defined expressions for the transformation of the basis under the action of the generators are written down

an analysis for the delta function with support on the light cone the authors study the delta function delta zeta zeta where zeta is a point in the minkowski space and quantity zeta zeta t2 x21 x22 x23 t2 r2 this generalised function has its support on the light cone t or r 0 which has an interesting singularity at its vertex 0 0 they present the derivatives of the multilayers spread over this cone their analysis is facilitated by the interplay between the theory of distributions generalised functions and the theory of singular surfaces

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

dimension formulas for the lie superalgebra sl m n although character formulas for simple finite dimensional modules of the lie superalgebra sl m n are not known in general they are known in the cases of so called typical and of singly atypical modules for singly atypical modules however the character formula is such that the classical limiting procedure yielding a dimension formula from a character formula does in general not work in this article another approach is presented yielding an explicit dimension formula for all singly atypical modules of sl m n in terms of the highest weight labels

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

some results in discontinuous fields and wave propogation abstract we present various applications of recently developed techniques which are an interplay between the differential geometry and the theory of distributions we apply them to the microlocal theory derive the general order transport equations for the strength of wave fronts obtain the values of the jumps of the general nth order derivatives of the harmonic functions across the potential layers and present the explicit formulas for an arbitrary order derivative of functions of the radial distance r

finite dimensional representations of the quantum superalgebra u q gl n m and related q identities explicit expressions for the generators of the quantum superalgebrau q gl n m acting on a class of irreducible representations are given the class under consideration consists of all essentially typical representations for these a gel fand zetlin basis is known the verification of the quantum superalgebra relations to be satisfied is shown to reduce to a set ofq number identities

diagram and superfield techniques in the classical superalgebras introduces the concept of graded permutation group in the analysis of tensor operators in the classical superalgebras for u m n and su m n irreducible tensor representations correspond to classes of young tableaux with definite graded symmetry type diagram techniques are given for kronecker products dimensions and branching rules the tensor techniques are complemented by the introduction of a superfluid formalism in which u m n and su m n act on polynomial functions over the appropriate superspace such superfluids may admit constraints a general superfield interpolates between the classes of young tableaux which correspond to particular types of constraint the tensor and superfield techniques are illustrated with case studies of su 2 1 and su m 1

cohomology of lie superalgebras m n and 2 2n we explicitly compute the first and second cohomology groups of the classical lie superalgebras mathfrak sl m n and mathfrak osp 2 2n with coefficients in the finite dimensional irreducible modules and the kac modules we also show that the second cohomology groups of these lie superalgebras with coefficients in the respective universal enveloping algebras under the adjoint action vanish the latter result in particular implies that the universal enveloping algebras rm u mathfrak sl m n and rm u mathfrak osp 2 2n do not admit any non trivial formal deformations of gerstenhaber type

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

the use of copula functions for predictive analysis of correlations between extreme storm tides abstract in this paper we present a method used in quantitative description of weakly predictable hydrological extreme events at inland sea investigations for correlations between variations of individual measuring points employing combined statistical methods were carried out as a main tool for this analysis we used a two dimensional copula function sensitive for correlated extreme effects additionally a new proposed methodology based on detrended fluctuations analysis dfa and anomalous diffusion ad was used for the prediction of negative and positive auto correlations and associated optimum choice of copula functions as a practical example we analysed maximum storm tides data recorded at five spatially separated places at the baltic sea for the analysis we used gumbel clayton and frank copula functions and introduced the reversed clayton copula the application of our research model is associated with modelling the risk of high storm tides and possible storm flooding