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

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

cluster formation in mesoscopic systems abstract graph theoretical approach is used to study cluster formation in mesocsopic systems appearance of these clusters is due to discrete resonances which are presented in the form of a multigraph with labeled edges this presentation allows to construct all non isomorphic clusters in a finite spectral domain and generate corresponding dynamical systems automatically results of mathematica implementation are given and two possible mechanisms of cluster destroying are discussed

transfer matrix renormalization group method for general markov random fields abstract we seek the numerical calculation of partition functions of general markov random fields mrfs on an infinitely long twisted cylindrical lattice by using the transfer matrix renormalization group tmrg method the tmrg is a variant of the density matrix renormalization group dmrg which automatically truncates the hilbert space so that the properties of large systems can be precisely calculated while the dimension of the renormalized hilbert space remains constant we apply the tmrg to the decimation of the fundamental transfer matrix that we have proposed previously for general mrfs instead of the standard s e scheme for tmrg we propose a new e s e scheme and propose an alternative method for selecting the renormalized basis specifically the new e s e scheme keeps those singular value decomposition svd components of the fundamental transfer matrix that are relevant for the perron state and truncates the other irrelevant ones results for the ising model show that our method exhibits very impressive accuracy under a rather restricted computational resource simulations for another four general mrfs demonstrate that our tmrg method is superior to the classical monte carlo method in accuracy computational speed and in the possibility of treating a much larger system

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

laminated wave turbulence generic algorithms iii the model of laminated wave turbulence presented recently unites both types of turbulent wave systems statistical wave turbulence introduced by kolmogorov and brought to the present form by numerous works of zakharov and his scientific school since nineteen sixties and discrete wave turbulence developed in the works of kartashova in nineteen nineties the main new feature described by this model is the following discrete effects do appear not only in the long wave part of the spectral domain corresponding to small wave numbers but all through the spectra thus putting forth a novel problem construction of fast algorithms for computations in integers of order 1012 and more in this paper we present a generic algorithm for polynomial dispersion functions and illustrate it by application to gravitational water waves and oceanic planetary waves

partitioning of ensembles of weakly interacting dispersing waves in resonators into disjoint classes abstract weakly non linear interactions of a finite number of waves in a bounded domain are considered it is evident that the synchronism conditions in this case turn into equations in integers due to boundedness of the domain a theorem about the partitioning of the set of vectors satisfying synchronism conditions into non intersecting classes is proved cases are found in which for a fixed vector k with integer coordinates there exist no vectors constituting a solution together with it examples are given of situations in which the said classes are infinite finite or empty methods of algebraic number theory have been used

representations of low rank orthosymplectic superalgebras by superfield techniques representations of lie superalgebras may be realised as functions superfields on graded manifolds suitably chosen onset spaces the technique is illustrated for osp 1 2 and osp 2 2 with a review of well known results and applied in the cases of osp 3 2 and osp 4 2 to construct some classes of irreducible representations

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

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

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

the gel fand basis and matrix elements of the graded unitary group u m n projection operators with correct normalisation are constructed for obtaining the gel fand basis mod nu m of the graded unitary group u m n each u m n gel fand basis vector mod nu m for an f particle system uniquely corresponds to a non standard basis vector mod nu m of the permutation group s f the matrix element of the generator ei 1i of u m n between the two gel fand basis vectors mod nu m and mod nu m is proportional to the overlap between the two non standard basis vectors mod nu m and mod nu m of s f explicit formulae are given for the normalisation constant in the projection operator as well as for the matrix elements of the generator ei 1i of the graded unitary group u m n which is the extension of the gel fand tsetlin formula for the ordinary unitary group u m

noisy spectra long correlations and intermittency in wave turbulence we study the k space fluctuations of the wave action about its mean spectrum in the turbulence of dispersive waves we use a minimal model based on the random phase approximation rpa and derive evolution equations for the arbitrary order one point moments of the wave intensity in the wave number space the first equation in this series is the familiar kinetic equation for the mean wave action spectrum whereas the second and higher equations describe the fluctuations about this mean spectrum the fluctuations exhibit a nontrivial dynamics if some long coordinate space correlations are present in the system as it is the case in typical numerical and laboratory experiments without such long range correlations the fluctuations are trivially fixed at their gaussian values and cannot evolve even if the wave field itself is non gaussian in the coordinate space unlike the previous approaches based on smooth initial k space cumulants the rpa model works even for extreme cases where the k space fluctuations are absent or very large and intermittent we show that any initial non gaussianity at small amplitudes propagates without change toward the high amplitudes at each fixed wave number at each fixed amplitude however the probability distribution function becomes gaussian at large time

a numerical study on statistical characteristics of amplitude fluctuations in a wave turbulence abstract a series of numerical studies has been performed to investigate the statistical characteristics of fluctuations of energy s a k 2 of each component wave and its time derivative s d a k 2 d t in a wave turbulence for a hamiltonian system with a non decay type dispersion relation it has been found that the probability density functions pdf of s and s evaluated among independent realizations are almost equivalent to the pdf s evaluated among nearby modes in the k space in a single realization therefore the averaging over the realizations and that over the nearby modes in a single realization are interchangeable it has also been found that the pdf of s evolves in time toward the 2 distribution with two degrees of freedom while that of s evolves toward the laplace distribution or the two sided exponential distribution at any wavenumber k both in the peripheral of the spectrum large k region and in the energy containing range small k region

on schur complements of sign regular matrices of order k abstract the issue regarding schur complements of sign regular matrices is rather subtle it is known that the class of totally nonnegative matrices is not closed under arbitrary schur complementation in this paper we demonstrate how schur complements of sign regular matrices of order k are sign regular of a certain order in particular some results for totally nonnegative and totally nonpositive matrices are provided as our corollaries

inequalities for permanents involving perron complements abstract let a r n n and let and be nonempty complementary subsets of 1 n of increasing integers for a we define the generalized perron complement of a in a at as the matrix p a a a a i a 1 a for the classes of the nonnegative matrices and of the positive semidefinite matrices we study the relationship between the permanents of the whole matrices and the permanents of their perron complement our conditions which hold in many cases of interest are such that the value of the permanent increases as we pass from the whole matrix to its generalized perron complement for nonnegative and irreducible matrices we also study the relationship between the maximum circuit geometric mean of the entire matrix and the maximum circuit geometric mean of its perron complements

discreteness and its effect on water wave turbulence abstract we perform numerical simulations of the dynamical equations for a free water surface in a finite basin in the presence of gravity wave turbulence wt is a theory derived for describing the statistics of weakly nonlinear waves in the infinite basin limit its formal applicability condition on the minimal size of the computational basin is impossible to satisfy in present numerical simulations and the number of wave resonances is significantly depleted due to the wavenumber discreteness the goal of this paper will be to examine which wt predictions survive in such discrete systems with depleted resonances and which properties arise specifically due to the discreteness effects as in a i dyachenko a o korotkevich v e zakharov weak turbulence of gravity waves jetp lett 77 10 2003 phys rev lett 92 13 2004 134501 m onorato et al freely decaying weak turbulence for sea surface gravity waves phys rev l 89 14 2002 n yokoyama statistics of gravity waves obtained by direct numerical simulation jfm 501 2004 169 178 our results for the wave spectrum agree with the zakharov filonenko spectrum predicted within wt we also go beyond finding the spectra and compute the probability density function pdf of the wave amplitudes and observe an anomalously large with respect to gaussian probability of strong waves which is consistent with recent theory y choi y v lvov s nazarenko b pokorni anomalous probability of large amplitudes in wave turbulence phys lett a 339 3 5 2004 361 369 also on arxiv math ph 0404022 v1 y choi y v lvov s nazarenko probability densities and preservation of randomness in wave turbulence phys lett a 332 2004 230 238 joint statistics of amplitudes and phases in wave turbulence physica d 201 2005 121 149 y choi y v lvov s nazarenko wave turbulence in recent developments in fluid dynamics vol 5 2004 transworld research network kepala india also on arxiv org math ph 0412045 using a simple model for quasi resonances we predict an effect arising purely due to discreteness the existence of a threshold wave intensity above which a turbulent cascade develops and proceeds to arbitrarily small scales numerically we observe that the energy cascade is very bursty in time and is somewhat similar to sporadic sandpile avalanches we explain this as a cycle a cascade arrest due to discreteness leads to accumulation of energy near the forcing scale which in turn leads to widening of the nonlinear resonance and therefore triggering of the cascade draining the turbulence levels and returning the system to the beginning of the cycle

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

classification of all star irreps of gl m n all the finite dimensional star irreps of gl m n are classified in terms of their highest weights thereby completing the classification of all finite dimensional star irreps of the basic classical lie superalgebras the lowest weights of such irreps are determined explicitly and it is shown that the contravariant and covariant tensor irreps of gl m n are star irreps of type 1 and 2 respectively this explains the applicability of the young diagram method to these two classes of representations however it is shown that one parameter families of typical star irreps occur that are intrinsically different from the tensorial irreps

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

on perron complements of inverse n0 matrices abstract in this paper we show that the perron complements of irreducible n 0 matrices are n 0 matrices we also demonstrate the perron complements of irreducible inverse n 0 matrices are inverse n 0 matrices with a little restriction in addition we give some related inequalities about inverse n 0 matrices

classification of all star and grade star irreps of gl n 1 a method for inducing nondegenerate forms on irreducible gl m n modules that implies some general results on star and grade star representations is investigated these results are applied to obtain a complete classification in terms of highest weights of the irreducible star and grade star representations of gl n 1 it is demonstrated that while gl n 1 admits a large class of star representations the irreducible grade star representations are comparatively rare moreover for n 2 all grade star irreducible representations are also star representations and for n 2 are atypical the superalgebra gl 2 1 proves to be a special case and admits a two parameter family of four dimensional typical grade star irreducible representations that are not star representations in particular typical grade star irreducible representations of gl n 1 exist only for n 1 2

free fermions and the alexander conway polynomial we show how the conway alexander polynomial arises from theq deformation of z2 graded sl n n algebras in the simplestsl 1 1 case we then establish connection between classical knot theory and its modern versions based on quantum groups we first shown how the crystal and the fundamental group of the complement of a knot give rise naturally to the burau representation of the braid group the burau matrix is then transformed into theu q sl 1 1 r matrix by going to the exterior power algebra using a det str identity this allows us to recover the state model of k2 89 as well we also show how theu q sl 1 1 algebra describes free fermions propagating on the knot diagram we rewrite the conway alexander polynomial as a berezin integral and thus as an apparently new determinant

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

on the structure of the observable algebra of qcd on the lattice we prove that the matter field subalgebra of the observable algebra for qed on a finite lattice is isomorphic to the enveloping algebra of the lie algebra sl 2n c factorized by a certain ideal using this result we give a new proof of the decomposition of the physical hilbert space into charge superselection sectors

the clebsch gordan coefficient for su m n gel fand basis the outer product reduction coefficients orc of the graded state permutation group which differ from the orc of the ordinary permutation group only in signs have been identified with the clebsch gordan coefficients cgc for the special gel fand basis of the graded unitary group su m n the cgc for a general gel fand basis of su m n can be easily obtained from those for the special one tables of the su m n cgc are presented which are valid for arbitrary m and n

on properties of weakly nonlinear wave interactions in resonators abstract weakly nonlinear wave interactions in resonators are considered qualitative differences from the continuous case are found prevalence of waves not taking part in any interactions at all dependence of the number of interacting waves on the basin form existence of basins where interactions are totally impossible locality of interactions in the spectral space consideration of equations defining the resonance surface in finite fields as well as methods of number theory are used some aspects of relations between the continuous and discrete cases are discussed

on the lie superalgebra embedding c n 1 b 0 n and dimension formulas it is shown that the orthosymplectic lie superalgebras osp m 2n have a nonregular subalgebra osp 1 2n this implies that paraboson operators can be realized as elements of osp m 2n the embedding osp 2 2n osp 1 2n or in a different notation c n 1 b 0 n is studied in more detail in particular branching rules are determined for all typical and atypical irreducible representations of osp 2 2n with respect to the subalgebra osp 1 2n finally dimension and superdimension formulas are given for the lie superalgebras under consideration

graded tensor calculus we develop a graded tensor calculus corresponding to arbitrary abelian groups of degrees and arbitrary commutation factors the standard basic constructions and definitions like tensor products spaces of multilinear mappings contractions symmetrization symmetric algebra as well as the transpose adjoint and trace of a linear mapping are generalized to the graded case and a multitude of canonical isomorphisms is presented moreover the graded versions of the classical lie algebras are introduced and some of their basic properties are described

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

uncoupling the perron eigenvector problem abstract for a nonnegative irreducible matrix a with spectral radius this paper is concerned with the determination of the unique normalized perron vector which satisfies a 62 0 j j 1 it is explained how to uncouple a large matrix a into two or more smaller matrices say p 11 p 22 p kk such that this sequence of smaller matrices has the following properties 1 each p ii is also nonnegative and irreducible so that each pii has a unique perron vector i 2 each pii has the same spectral radius as a 3 it is possible to determine the i s completely independently of each other so that one can execute the computation of the i s parallel 4 it is easy to couple the smaller perron vectors i back together in order to produce the perron vector for the original matrix a

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

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

analytic bethe ansatz related to a one parameter family of finite dimensional representations of the lie superalgebra as is well known the type 1 lie superalgebra admits a one parameter family of finite dimensional irreducible representations we have carried out an analytic bethe ansatz related to this family of representations we present formulae which are deformations of previously proposed determinant formulae labelled by a young superdiagram these formulae will provide a transfer matrix eigenvalue in a dressed vacuum form related to the solutions of a graded yang baxter equation which depend not only on the spectral parameter but also on a non additive continuous parameter a class of transfer matrix functional relations among these formulae is briefly mentioned

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

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

on perron complements of totally nonnegative matrices abstract an n n matrix is called totally nonnegative if every minor of a is nonnegative the problem of interest is to describe the perron complement of a principal submatrix of an irreducible totally nonnegative matrix we show that the perron complement of a totally nonnegative matrix is totally nonnegative only if the complementary index set is based on consecutive indices we also demonstrate a quotient formula for perron complements analogous to the so called quotient formula for schur complements and verify an ordering between the perron complement and schur complement of totally nonnegative matrices when the perron complement is totally nonnegative

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

the clebsch gordan coefficients and isoscalar factors of the graded unitary group su m n it is pointed out that the clebsch gordan coefficients cgc of the permutation group are the coefficients which couple irreducible bases irb of su m n and su p q to the irb of su mp nq mp np and the outer product reduction coefficients orc of the permutation group are the coefficients which couple the irb of su m n and su p q into the irb of su m p n q the orc of permutation groups with due modification in signs are the cgc for the special gel fand basis of su m n the isoscalar factor isf for the permutation group chain s f contains implies s f1 s f2 is the isf for the graded unitary group chain su mp nq mq np contains implies su m n su p q and the outer product isf for the group chain s f contains implies s f1 s f2 is the isf for su mp nq mq np contains implies su m n x su p q all these coefficients can be calculated easily and some are already available

the schur complement of ph matrices abstract in this paper we consider the class of p h matrices a subclass of h matrices and using scaling characterization we show that this class is closed under taking the schur complement we show that under certain conditions the perron complement of a p h matrix is a p h matrix we also present a way of constructing a scaling matrix for the given p h matrix and we give eigenvalue localization for the schur complement of a p h matrix using only the entries of the original matrix we illustrate this by numerical examples

high phase lag order runge kutta and nystrom pairs we exploit the freedom in the selection of the free parameters of one family of eighth algebraic order runge kutta rk pairs and of three families of fourth sixth and eighth order rk nystrom rkn pairs with the purpose of obtaining specific pairs of the highest possible phase lag order which are also characterized by minimized principal truncation error coefficients we present a method for the analytic derivation of the dissipation order conditions for rk methods and the phase lag and dissipation order conditions for nystrom methods the rk pairs we study here are based on a one parameter generalization of some older families of pairs an algorithm and specific optimized 8 6 nystrom pairs are also provided for a class of initial value problems whose solution is known to be described by free oscillations or free oscillations of low frequency with forced oscillations of high frequency superimposed over long integration intervals these new pairs seem to offer some advantages with respect to some older pairs the latter are of the same algebraic orders as the new ones but are characterized by the minimal phase lag order according to their algebraic order and number of stages

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

perron eigenvector of the tsetlin matrix abstract we consider the move to position k linear search scheme where the sequence of record requests is a markov chain formulas are derived for the stationary distribution of the permutation chain for k 1 2 n 1 and n where n is the number of records certain identities for the perron complement are established in the process

inverses of perron complements of inverse m matrices the concept of the perron complement of a nonnegative and irreducible matrix was introduced by meyer in 1989 and it was used by him to construct an algorithm for computing the stationary distribution vector for markov chains here we consider properties of the perron complement of an n n matrix k which is an inverse of an irreducible m matrix we first show that the perron complements of k are inverses of m matrices and that the inverses of associated principal submatrices of k are sandwiched between the inverses of the perron complements of k and the inverses of the corresponding schur complements of k we then investigate the directed graph of the inverse of the perron complements of such matrices k

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

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

generalised verma modules for the orthosymplectic lie superalgebra ospk 2 abstract the composition factors with their multiplicities and the submodule lattices are determined for generalised verma modules over the orthosymplectic lie superalgebra osp k 2 the results enable us to obtain explicit formulae for the formal characters and dimensions of the finite dimensional irreducible modules applying these results we also compute the first and second cohomology groups of the lie superalgebra with coefficients in finite dimensional kac modules and irreducible modules

asymptotics of the partition function of a general markov random field on an infinite rectangular lattice abstract we investigate theoretically and numerically the asymptotics of the partition function of a general markov random field mrf on an infinite rectangular lattice we first propose the general local energy function lef parameterized mrf then we prove that the thermodynamic limit of the free energy of the mrf can be exactly characterized by the perron root of the fundamental transfer matrix of a particular markov additive process map this matrix possesses a special structure and many interesting properties that enable parallel computation of the perron root and may be beneficial for deriving an analytical form of the free energy we also develop another transfer matrix for numerical computation of the desired perron root specifically the former is a site to site transfer matrix on a twisted cylindrical lattice while the latter is the one associated with a row to row transition on a vertical strip numerical results show that our methods exhibit consistent finite size scaling behavior even for small values of the lattice width this study reveals that the fundamental transfer matrix is an alternative direction of research on the analysis of the partition function of general mrfs within the scope of matrix algebra

the casimir invariants and gel fand basis of the graded unitary group su m n in the space of f particle product states the casimir invariants of the graded unitary group su m n are functions of the class operators of the graded coordinate permutation group s f which are equal to the class operators of the graded state permutation group s f formulae for the quadratic and cubic casimir invariants are given it is shown that in general the eigenvalue formulae for the u m n casimir invariants as functions of the partition can be obtained from those for u m n by simply changing the argument of the functions from m n to m n the quasi standard basis of s f is identified with the gel fand basis of su m n i e the irreducible basis classified according to the group chain su m n contains implies su m n 1 contains implies su m contains implies su 2 contains implies u 1 the special gel fand basis of su m n for which the weight is restricted to 1 1 1 can be easily obtained from the gel fand basis of su m n with due allowances for modifying sign factors arising from the grading the construction of a general su m n gel fand basis is also discussed

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

casimir invariants characteristic identities and tensor operators for strange superalgebras we define a class of super subalgebras of gl m n realized as the set of fixed points of a graded endomorphism of gl m n this class includes the superalgebras gp m and gq m related to the so called strange simple superalgebras p m and q m as well as osp m n general covariant contravariant and mixed tensor operators are defined for this class in terms of appropriate module homomorphisms traces of certain tensors give the usual sequence of casimir invariants for gp m these are shown to vanish identically while for gq m eigenvalues of the quadratic and cubic casimir invariants are derived in terms of highest weights and a polynomial characteristic identity is exhibited

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

negative dimensional groups in quantum physics the author shows that an explicit definition of negative dimensional classical groups in terms of their action on grassmann representation spaces together with weyl s character formula for the grassmann tensorial representations of such groups leads to a framework in which some surprising negative dimensional properties of group theoretic invariants arise naturally as an application he shows that the spectra of the grassmann harmonic oscillator and of grassmann angular momentum are related to their bosonic counterparts by a simple analytic continuation in the dimension

stochastic complementation uncoupling markov chains and the theory of nearly reducible systems a concept called stochastic complementation is an idea which occurs naturally although not always explicitly in the theory and application of finite markov chains this paper brings this idea to the forefront with an explicit definition and a development of some of its properties applications of stochastic complementation are explored with respect to problems involving uncoupling procedures in the theory of markov chains furthermore the role of stochastic complementation in the development of the classical simon ando theory of nearly reducible system is presented

classification of finite dimensional unitary irreps for uq gl m n all finite dimensional irreducible unitary representations of the quantum supergroup uq gl m n are classified in terms of their highest weights for generic q 0

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

casimir invariants characteristic identities and young diagrams for color algebras and superalgebras the generalized commutation relations satisfied by generators of the general linear special linear and orthosymplectic color super algebras are presented in matrix form tensor operators including casimir invariants are constructed in the enveloping algebra for the general special linear and orthosymplectic cases eigenvalues of the quadratic and higher casimir invariants are given in terms of the highest weight vector correspondingly characteristic polynomial identities satisfied by the matrix of generators are obtained in factorized form classes of finite dimensional representations are identified using young diagram techniques and dimension branching and product rules for these are given finally the connection between color super algebras and generalized particle statistics is elucidated

a character formula for atypical critical gl m n representations labelled by composite partitions let be the lie superalgebra we show how to associate a weight to a composite partition with composite young diagram based upon the definition of critical representations the notion of critical composite partition is introduced it is shown that for critical composite partitions subject to a technical restriction the corresponding representation v is tame so its character formula can be computed this character is shown to coincide with the composite s function

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

total nonnegativity of the extended perron complement abstract a real matrix is called totally nonnegative if all of its minors are nonnegative in this paper the extended perron complement of a principal submatrix in a matrix a is investigated in extension of known results it is shown that if a is irreducible and totally nonnegative and the principal submatrix consists of some specified consecutive rows then the extended perron complement is totally nonnegative also inequalities between minors of the extended perron complement and the schur complement are presented

hook young diagrams with applications to combinatorics and to representations of lie superalgebras definition et etude des tableaux k l semi standards applications a la combinatoire et aux representations des superalgebres de lie des diagrammes de young a crochets