embeddings of schur functions into types b c d we consider the problem of embedding the semi ring of schur positive symmetric polynomials into its analogue for the classical types b c d if we preserve highest weights and add the additional lie theoretic parity assumption that the weights in images of schur functions lie in a single translate of the root lattice there are exactly two solutions these naturally extend the kirillov reshetikhin decompositions of representations of symplectic and orthogonal quantum affine algebras uq g

bases for some reciprocity algebras ii abstract we construct bases for the stable branching algebras for the symmetric pairs gl n o n o n m o n o m and sp 2 n gl n

combinatorial r matrices for a family of crystals b 1 n d 1 n a 2 2n and d 2 n 1 cases abstract for coherent families of crystals of affine lie algebras of type b 1 n d 1 n a 2 2 n and d 2 n 1 we describe the combinatorial r matrix using column insertion algorithms for b c d young tableaux this is a continuation of previous work by the authors 2000 in physical combinatorics m kashiwara and t miwa eds birkhauser boston

langlands duality for representations of quantum groups we establish a correspondence or duality between the characters and the crystal bases of finite dimensional representations of quantum groups associated to langlands dual semi simple lie algebras this duality may also be stated purely in terms of semi simple lie algebras to explain this duality we introduce an interpolating quantum group depending on two parameters which interpolates between a quantum group and its langlands dual we construct examples of its representations depending on two parameters which interpolate between representations of two langlands dual quantum groups

synthesis of modular contiguously clustered linear arrays through a sparseness regularized solver an innovative methodology for the design of modular and contiguously clustered linear arrays is proposed the approach formulates the subarraying problem as a pattern matching one in which elementary modules taken from a user defined dictionary are combined to synthesize the final antenna layout by recasting such a synthesis problem within the compressive sensing framework an efficient solver is used to determine the array architecture starting from the a priori specification of the features e g the length of the admissible subarrays a set of representative numerical experiments is discussed to assess the advantages and drawbacks of the proposed algorithm also in comparison with some state of the art methods

a simple algorithm to achieve desired patterns for arbitrary arrays a simple iterative algorithm which can be used to find array weights that produce array patterns with a given look direction and an arbitrary sidelobe specification is presented the method can be applied to nonuniform array geometries in which the individual elements have arbitrary and differing radiation patterns the method is iterative and uses sequential updating to ensure that peak sidelobe levels in the array meet the specification computation of each successive pattern is based on the solution of a linearly constrained least squares problem the constraints ensure that the magnitude of the sidelobes at the locations of the previous peaks takes on the prespecified values phase values for the sidelobes do not change during this process and problems associated with choosing a specific phase value are therefore avoided experimental evidence suggests that the procedure terminates in remarkably few iterations even for arrays with significant numbers of elements

on the decomposition of tensor products of the representations of the classical groups by means of the universal characters on definit les caracteres universels d une representation rationnelle de gl n c et on donne les formules de decomposition des produits tensoriels entre eux en termes de coefficients de littlewood richardson on demontre les formules de decomposition du produit tensoriel de deux representations irreductibles de sp 2n c et de so n c

analytic bethe ansatz and functional equations associated with any simple root systems of the lie superalgebra sl r 1 s 1 abstract the lie superalgebra sl r 1 s 1 admits several inequivalent choices of simple root systems we have carried out analytic bethe ansatz for any simple root systems of sl r 1 s 1 we present transfer matrix eigenvalue formulae in dressed vacuum form which are expressed as the young supertableaux with some semistandard like conditions these formulae have determinant expressions which can be viewed as quantum analogue of jacobi trudi and giambelli formulae for sl r 1 s 1 we also propose a class of transfer matrix functional relations which is specialization of hirota bilinear difference equation using the particle hole transformation relations among the bethe ansatz equations for various kinds of simple root systems are discussed

x m for symmetric powers author s schilling anne shimozono mark abstract the x m conjecture of hatayama et al asserts the equality between the one dimensional configuration sum x expressed as the generating function of crystal paths with energy statistics and the fermionic formula m for all affine kac moody algebra in this paper we prove the x m conjecture for tensor products of kirillov reshetikhin crystals b 1 s associated to symmetric powers for all nonexceptional affine algebras

combinatorial structure of kirillov reshetikhin crystals of type dn 1 bn 1 a2n 1 2 abstract we provide the explicit combinatorial structure of the kirillov reshetikhin crystals b r s of type d n 1 b n 1 and a 2 n 1 2 this is achieved by constructing the crystal analogue of the automorphism of the d n 1 respectively b n 1 or a 2 n 1 2 dynkin diagram that interchanges the 0 and 1 node the involution is defined in terms of new diagrams that govern the d n to d n 1 respectively b n to b n 1 or c n to c n 1 branching it is also shown that the crystal b r s is perfect these crystals have been implemented in mupad combinat the implementation is discussed in terms of many examples

quantization of branching coefficients for classical lie groups abstract we study natural quantizations k of branching coefficients corresponding to the restrictions of the classical lie groups to the levi subgroups of their standard parabolic subgroups the polynomials obtained can be regarded as generalizations of lusztig q analogues of weight multiplicities for gl n they coincide with poincare polynomials previously studied by shimozono and weyman they also appear in the hilbert series of the euler characteristic of certain graded virtual g modules and by a result of broer admit nonnegative coefficients providing that restrictive conditions are verified when the levi subgroup considered is isomorphic to a direct product of linear groups we prove that these polynomials admit a stable limit k which decomposes as nonnegative integer combination of poincare polynomials for a general levi subgroup it is conjectured that the polynomials k have nonnegative coefficients when they are parametrized by two partitions when g gl n the polynomials k can be interpreted as quantizations of the littlewood richardson coefficients we show that there also exists a duality between tensor product coefficients for types b c d defined as the analogues of the littlewood richardson coefficients and branching coefficients corresponding to the restriction of so 2 n to subgroups defined from orthogonal decompositions of the root system d n which are not levi subgroups these coefficients can also be quantified but the q analogues obtained may have negative coefficients given a tensor product of irreducible gl n modules we then introduce for each classical group g so n or sp n some q analogues d of the multiplicities obtained by decomposing into its g irreducible components we establish a duality between the polynomials d and k according to a conjecture by shimozono the stable one dimensional sums for nonexceptional affine crystals are expected to coincide with the polynomials d associated to a sequence of rectangular partitions of decreasing heights

crystal energy functions via the charge in types a and c the ram yip formula for macdonald polynomials at t 0 provides a statistic which we call charge in types a and c it can be defined on tensor products of kashiwara nakashima single column crystals in this paper we prove that the charge is equal to the negative of the energy function on affine crystals the algorithm for computing charge is much simpler and can be more efficiently computed than the recursive definition of energy in terms of the combinatorial r matrix

parabolic kazhdan lusztig polynomials and schubert varieties abstract we shall give a description of the intersection cohomology groups of the schubert varieties in partial flag manifolds over symmetrizable kac moody lie algebras in terms of parabolic kazhdan lusztig polynomials introduced by deodhar

affine structures and a tableau model for e_6 crystals abstract we provide the unique affine crystal structure for type e 6 1 kirillov reshetikhin crystals corresponding to the multiples of fundamental weights s 1 s 2 and s 6 for all s 1 in bourbaki s labeling of the dynkin nodes where 2 is the adjoint node our methods introduce a generalized tableaux model for classical highest weight crystals of type e and use the order three automorphism of the affine e 6 1 dynkin diagram in addition we provide a conjecture for the affine crystal structure of type e 7 1 kirillov reshetikhin crystals corresponding to the adjoint node

combinatorics of regular a2 crystals abstract we show that a connected regular a 2 crystal the crystal graph of a highest weight integrable module over u q s l 3 can be produced from two half grids by replicating them and gluing together in a certain way also some extensions and related aspects are discussed

young diagrammatic methods for the representation theory of the classical groups of type bn cn dn since the appearance of the celebrated article the classical groups of weyl 7 numerous influences have occurred in the representation theory of the classical groups especially it has become folklore that the facts in this field should be expressed only in terms of young diagrams the aim of this paper is one of the realizations of this principle a classical model of our approach is the results for gl n which are known as littlewood richardson coefficients and the kostka coefficients these two coefficients can be stated in the combinatorial manner using only young diagrams independently of the rank n of gl n we generalize these results to the other classical groups over c namely we give explicit formulas to decompose the tensor product of two irreducible representations into irreducible constituents also we have obtained an interesting relationship linking the irreducible characters of sp and so which we call duality relation between the representations of sp and so as for explicit formulas for the multiplicities of weights in the form of polynomials in the rank n we refer to 3

affine approach to quantum schubert calculus this paper presents a formula for products of schubert classes in the quantum cohomology ring of the grassmannian we introduce a generalization of schur symmetric polynomials for shapes that are naturally embedded in a torus then we show that the coefficients in the expansion of these toric schur polynomials in terms of the regular schur polynomials are exactly the 3 point gromov witten invariants which are the structure constants of the quantum cohomology ring this construction implies three symmetries of the gromov witten invariants of the grassmannian with respect to the groups s3 z nz 2 and z 2z the last symmetry is a certain curious duality of the quantum cohomology which inverts the quantum parameter q our construction gives a solution to a problem posed by fulton and woodward about the characterization of the powers of the quantum parameter q which occur with nonzero coefficients in the quantum product of two schubert classes the curious duality switches the smallest such power of q with the highest power we also discuss the affine nil temperley lieb algebra that gives a model for the quantum cohomology

pattern synthesis with multipoint accurate array response control in this paper the problem of pattern synthesis with antenna arrays is addressed and new approaches based on the recently developed accurate array response control textrm a 2 textrm rc algorithm are presented it is shown that the array weight vector obtained by the textrm a 2 textrm rc algorithm to control the normalized response at a single direction in each step belongs to a specific set thus an appropriate weight vector chosen from the intersection of weight vector sets corresponding to the desired responses at multiple directions is capable of simultaneously controlling those responses this results in the so called multipoint accurate array response control textrm ma 2 textrm rc algorithm moreover in order to avoid possible beam axis shift in pattern synthesis a modified textrm ma 2 textrm rc textrm m 2 textrm a 2 textrm rc algorithm is proposed by imposing a derivative constraint on the direction of beam axis representative numerical examples are provided to demonstrate the effectiveness of the proposed textrm ma 2 textrm rc and textrm m 2 textrm a 2 textrm rc algorithms for multipoint responses control and pattern synthesis

maximum entropy estimation of density and regression functions abstract this paper presents an information efficient technique to determine the functional forms of density and regression functions by maximizing entropy subject to given side conditions a flexible functional form approach is established as special cases of this approach exponential polynomial cobb douglas translog generalized cobb douglas generalized leontief fourier flexible form and minflex laurent series etc can be derived in a monte carlo experiment the maximum entropy regression method shows good performance in terms of goodness of fit boxplots and isoquants produced by the maximum entropy method are compared with those produced by other well known functions

schensted type correspondence plactic monoid and jeu de taquin for type cn we use kashiwara s theory of crystal bases to study the plactic monoid for uq sp2n then we describe the corresponding insertion and sliding algorithms the sliding algorithm is essentially the symplectic jeu de taquin defined by sheats and our construction gives the proof of its compatibility with plactic relations

a flexible array synthesis method using quadratic programming a highly flexible synthesis method for an arbitrary array is proposed to best approximate a desired array pattern in a minimum mean square error sense the basic idea of the technique is to form a quadratic program with its cost function given by the mean square error between the array response and a properly selected pattern described by a known mathematical function this quadratic program can be a constrained or unconstrained optimization problem depending on the requirements of the desired array pattern in formulating the quadratic program no assumption has been made on the gain phase response or characteristics of the individual array elements therefore one can synthesize an array of arbitrary shape to any appropriate pattern with the characteristic of the array elements taken into consideration as long as one is able to model the array accurately the proposed method is used to synthesize arrays of different shapes linear as well as planar arrays including rectangular and circular planar arrays using a chebyshev polynomial or zero function as a design template to illustrate the feasibility of the proposed method

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

stable rigged configurations for quantum affine algebras of nonexceptional types for an affine algebra of nonexceptional type in the large rank we show the fermionic formula depends only on the attachment of the node 0 of the dynkin diagram to the rest and the fermionic formula of not type a can be expressed as a sum of that of type a with littlewood richardson coefficients combining this result with kirillov et al 2002 13 and lecouvey et al 2011 18 we settle the x m conjecture under the large rank hypothesis

affine crystals one dimensional sums and parabolic lusztig q analogues this paper is concerned with one dimensional sums in classical affine types we prove a conjecture of shimozono and zabrocki j algebra 299 33 61 2006 by showing they all decompose in terms of one dimensional sums related to affine type a provided the rank of the root system considered is sufficiently large as a consequence any one dimensional sum associated to a classical affine root system with sufficiently large rank can be regarded as a parabolic lusztig q analogue

a maximum entropy approach to recovering information from multinomial response data abstract the classical maximum entropy me approach to estimating the unknown parameters of a multinomial discrete choice problem which is equivalent to the maximum likelihood multinomial logit ml estimator is generalized the generalized maximum entropy gme model includes noise terms in the multinomial information constraints each noise term is modeled as the mean of a finite set of a priori known points in the interval 1 1 with unknown probabilities where no parametric assumptions about the error distribution are made a gme model for the multinomial probabilities and for the distributions associated with the noise terms is derived by maximizing the joint entropy of multinomial and noise distributions under the assumption of independence the gme formulation reduces to the me in the limit as the sample grows large or when no noise is included in the entropy maximization further even though the gme and the logit estimators are conceptually different the dual gme model is related to a gener

crystals and trees quasi kashiwara operators monoids of binary trees and robinson schensted type correspondences abstract kashiwara s crystal graphs have a natural monoid structure that arises by identifying words labelling vertices that appear in the same position of isomorphic components the celebrated plactic monoid the monoid of young tableaux arises in this way from the crystal graph for the q analogue of the general linear lie algebra gl n and the so called kashiwara operators interact beautifully with the combinatorics of young tableaux and with the robinson schensted knuth correspondence the authors previously constructed an analogous quasi crystal structure for the related hypoplactic monoid the monoid of quasi ribbon tableaux which has similarly neat combinatorial properties this paper constructs an analogous crystal type structure for the sylvester and baxter monoids the monoids of binary search trees and pairs of twin binary search trees respectively both monoids are shown to arise from this structure just as the plactic monoid does from the usual crystal graph the interaction of the structure with the sylvester and baxter versions of the robinson schensted knuth correspondence is studied the structure is then applied to prove results on the number of factorizations of elements of these monoids and to prove that both monoids satisfy non trivial identities

extended nakajima monomials and crystal b for finite simple lie algebras abstract nakajima introduced a certain set of monomials realizing the irreducible highest weight crystals b the monomial set can be extended so that it contains the crystal b in addition to b we study the crystal base b of the negative part of a quantum group for finite types b n c n d n and g 2 we present new descriptions of the crystal b in the language of extended nakajima monomials there is a natural correspondence between the monomial description and young tableau description which is another description of b

pattern characteristics of linear arrays using the constrained least squares distribution the constrained least squares cls distribution is a method for obtaining distribution functions that yield low sidelobe patterns with specified constraints on the aperture efficiency and are especially useful for the transmit patterns of active array antennas the widely used taylor distribution optimizes only pattern performance while the cls distribution optimizes pattern performance while taking into account the constraints on both the peak element amplitude and the total effective radiated voltage erv of the aperture distribution the paper compares the pattern characteristics of linear arrays with cls and taylor distributions the results help to establish guidelines on when a cls distribution would be preferable over a taylor distribution when a specified aperture efficiency is important

a numerical pattern synthesis algorithm for arrays a numerical technique for pattern synthesis in arrays is presented for a given set of elements the technique allows one to find a set of array coefficients that steer the main beam in a given direction and yield sidelobes meeting a specified criterion if such a set of array coefficients exists if the pattern specifications cannot be met with the given elements the algorithm finds the best attainable pattern the advantage of this technique is that it can be used with an arbitrary set of array elements different elements in the array can have different element patterns and the array can have arbitrary nonuniform spacing between elements the synthesis technique is based on adaptive array theory the given array elements are assumed to be used as the elements of an adaptive array the main beam is pointed in the proper direction by choosing the steering vector for that direction and the sidelobes are controlled by introducing a large number of interfering signals at many angles throughout the sidelobe region the algorithm iterates on the interference powers until a suitable pattern is obtained

geometric crystals on unipotent groups and generalized young tableaux abstract we define geometric unipotent crystal structure on maximal unipotent subgroups of semi simple algebraic groups we shall show that in a n case their ultra discretizations coincide with crystals obtained by generalizing young tableaux

numerical synthesis of arbitrary discrete arrays a numerical algorithm for the synthesis of arbitrary discrete arrays is given the technique is based on the algorithm of c a olen and r t compton 1990 but employs a different iterative recursive algorithm yielding an optimum convergence rate it is shown that the new algorithm converges to the optimum solution in a significantly reduced number of iterations compared with the technique of olen and compton use of the technique is illustrated by applying the synthesis procedure to a ten element linear array and to a conformal array consisting of ten dipoles located on a parabolic curve

the octahedron recurrence and gln crystals abstract we study the hive model of g l n tensor products following knutson tao and woodward we define a coboundary category where the tensor product is given by hives and where the associator and commutor are defined using a modified octahedron recurrence we then prove that this category is equivalent to the category of crystals for the lie algebra g l n the proof of this equivalence uses a new connection between the octahedron recurrence and the jeu de taquin and schutzenberger involution procedures on young tableaux

odd symplectic groups on definit des groupes symplectiques sur des espaces vectoriels de dimension impaire ces groupes de lie ne sont ni simples ni reductifs on obtient une formule de caracteres pour certaines representations tensorielles de ces groupes

taylor patterns for discrete arrays the equivalent of a taylor pattern for continuous apertures is developed for discrete arrays this array factor can be realized exactly since it belongs to the appropriate class of patterns a method for determining the element excitations is also presented as an example the pattern and excitations are calculated for a 41 element array with bar n 6 and a 25 db design sidelobe level comparison is made with the patterns obtained by applying root matching and sampling of a continuous taylor aperture

bitableaux bases of the quantum coordinate algebra of a semisimple group abstract we extend the standard basis theorem of rota et al to the setting of quantum symmetrizable kac moody algebras in particular we obtain a procedure to give a presentation of the quantum coordinate algebra of any semisimple group for generic q more precisely given any integrable module v of a quantum symmetrizable kac moody algebra u q g we obtain a generating set of the ideal of relations among the matrix coefficients of v and we give an upper bound for the degrees of these polynomials our approach is based on the theory of crystal bases and littelmann s generalization of the plactic algebra

rodrigues formulas for the macdonald polynomials we present formulas of rodrigues type giving the macdonald polynomials for arbitrary partitions through the repeated application of creation operatorsbk k 1 l on the constant 1 three expressions for the creation operators are derived one from the other when the last of these expressions is used the associated rodrigues formula readily implies the integrality of the q t kostka coefficients the proofs given in this paper rely on the connection between affine hecke algebras and macdonald polynomials

twisted quantum affine algebras we give a highest weight classification of the finite dimensional irreducible representations of twisted quantum affine algebras as in the untwisted case such representations are in one to one correspondence with n tuples of monic polynomials in one variable but whereas in the untwisted case n is the rank of the underlying finite dimensional complex simple lie algebra in the twisted case n is the rank of the subalgebra of fixed by the diagram automorphism the way in which such an n tuple determines a representation is also more complicated than in the untwisted case

hall littlewood vertex operators and generalized kostka polynomials abstract a family of vertex operators that generalizes those given by jing for the hall littlewood symmetric functions is presented these operators produce symmetric functions related to the poincare polynomials referred to as generalized kostka polynomials in the same way that jing s operator produces symmetric functions related to kostka foulkes polynomials these operators are then used to derive commutation relations and new relations involving the generalized kostka coefficients such relations may be interpreted as identities in the gl n c equivariant k theory of the nullcone

lusztig s q analogue of weight multiplicity and one dimensional sums for affine root systems abstract in this paper we complete the proof of the x k conjecture that for every family of nonexceptional affine algebras the graded multiplicities of tensor products of symmetric power kirillov reshetikhin modules known as one dimensional sums have a large rank stable limit x that has a simple expression called the k polynomial as nonnegative integer combination of kostka foulkes polynomials we consider a subfamily of lusztig s q analogues of weight multiplicity which we call stable kl polynomials and denote by kl we give a type independent proof that k kl this proves that x kl the family of stable one dimensional sums coincides with family of stable kl polynomials our result generalizes the theorem of nakayashiki and yamada which establishes the above equality in the case of one dimensional sums of affine type a and the lusztig q analogue of type a where both are kostka foulkes polynomials

the structure of us food demand abstract an exactly aggregable system of gorman engel curves for us food consumption is developed and implemented box cox transformations on prices and income nest functional form the model nests rank up to rank three the model is estimated by nonlinear three stage least squares with annual time series data on 21 foods 17 nutrients age and race demographics and the distribution of income for 1919 1941 and 1947 2000 results are consistent with full rank three point estimates for the box cox parameters on income and prices are 0 86 and 1 09 respectively strongly rejecting zero and one in both cases no statistical evidence of serial correlation specification errors or parameter instability is found

analysis of a particle antiparticle description of a soliton cellular automaton we present a derivation of a formula that gives dynamics of an integrable cellular automation associated with crystal bases this automaton is related to type d affine lie algebra and contains usual box ball systems as a special case the dynamics is described by means of such objects as carriers particles and antiparticles we derive it from an analysis of a recently obtained formula of the combinatorial r an intertwiner between tensor products of crystals that was found in a study of geometric crystals

an effective approach for the optimal focusing of array fields subject to arbitrary upper bounds a new approach to the optimal focusing of array fields subject to arbitrary upper bounds is presented the approach formulates the problem as the minimization of a linear function in a convex set unlike other approaches this one guarantees the achievement of the global optimum by using local optimization techniques and can moreover deal with any convex constraint on the unknowns such as near field constraints optimization is performed by two ad hoc developed solution algorithms which exploit the geometrical characteristics of the problem at hand thus leading to extremely effective and computationally efficient numerical codes an extensive numerical analysis has been performed in all cases of linear planar and circular arc arrays the enhanced performance of the proposed technique with respect to the solution algorithms available in the literature fully confirms the effectiveness of the approach

ribbon tableaux hall littlewood functions quantum affine algebras and unipotent varieties we introduce a new family of symmetric functions which are q analogs of products of schur functions defined in terms of ribbon tableaux these functions can be interpreted in terms of the fock space representation fq of uq sln and are related to hall littlewood functions via the geometry of flag varieties we present a series of conjectures and prove them in special cases the essential step in proving that these functions are actually symmetric consists in the calculation of a basis of highest weight vectors of fq using ribbon tableaux

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

maximum entropy in the problem of moments the maximum entropy approach to the solution of underdetermined inverse problems is studied in detail in the context of the classical moment problem in important special cases such as the hausdorff moment problem we establish necessary and sufficient conditions for the existence of a maximum entropy solution and examine the convergence of the resulting sequence of approximations a number of explicit illustrations are presented in addition to some elementary examples we analyze the maximum entropy reconstruction of the density of states in harmonic solids and of dynamic correlation functions in quantum spin systems we also briefly indicate possible applications to the lee yang theory of ising models to the summation of divergent series and so on the general conclusion is that maximum entropy provides a valuable approximation scheme a serious competitor of traditional pade like procedures

random walks in weyl chambers and crystals we use kashiwara crystal basis theory to associate a random walk w to each irreducible representation v of a simple lie algebra this is achieved by endowing the crystal attached to v with a possibly non uniform probability distribution compatible with its weight graduation we then prove that the generalized pitmann transform defined by biane bougerol and o connell for similar random walks with uniform distributions yields yet a markov chain when the representation is minuscule and the associated random walk has a drift in the weyl chamber we establish that this markov chain has the same law as w conditionned to never exit the cone of dominant weights at the heart of our proof is a quotient version of a renewal theorem that we state in the context of general random walks in a lattice

a new young diagrammatic method for kronecker products of and a new simple young diagrammatic method for kronecker products of o n and sp 2m is proposed based on the representation theory of brauer algebras a general procedure for the decomposition of the tensor products of representations for o n and sp 2m is outlined which is similar to that for u n known as the littlewood rules together with trace contractions from a brauer algebra and some modification rules given by king

information theoretic measures of the income distribution in food demand a new method to nest estimate and test the rank and functional form of the income terms in an incomplete system of demand equations is developed information theory is employed to infer the u s income distribution from data on quintile and top five percentile income ranges and intra quintile and top five percentile mean incomes maximum entropy income distributions are combined with data on the u s demand for 21 food items to estimate u s food demand over the period 1919 1995 excluding 1942 1946

stable branching rules for classical symmetric pairs we approach the problem of obtaining branching rules from the point of view of dual reductive pairs specifically we obtain a stable branching rule for each of 10 classical families of symmetric pairs in each case the branching multiplicities are expressed in terms of littlewood richardson coefficients some of the formulas are classical and include for example littlewood s restriction rule as a special case

explicit constructions of the fundamental representations of the symplectic lie algebras abstract we give two constructions for each fundamental representation of sp 2n c we also present quantum versions of these constructions these are explicit in the sense of the gelfand tsetlin constructions of the irreducible representations of gl n c we explicitly specify the matrix elements for certain generators of sp 2n c with respect to each of the two explicit bases presented in fact our constructions appear to have been the first such infinite family of explicit constructions of irreducible representations of simple lie algebras since the gelfand tsetlin constructions were obtained in 1950 our approach is combinatorial the key idea is to find a suitable family of partially ordered sets on which to present the action of the lie algebra and then to use these posets to produce the bases and the actions of the generators our constructions of the fundamental representations of sp 2n c take place on two families of posets which we call symplectic lattices previously 1999 j combin theory ser a88 217 234 we used these representation constructions to confirm a conjecture of reiner and stanton concerning one of these families of lattices

deformed universal characters for classical and affine algebras abstract creation operators are given for the three distinguished bases of the type bcd universal character ring of koike and terada yielding an elegant way of treating computations for all three types in a unified manner deformed versions of these operators create symmetric function bases whose expansion in the universal character basis has polynomial coefficients in q with nonnegative integer coefficients we conjecture that these polynomials are one dimensional sums associated with crystal bases of finite dimensional modules over quantized affine algebras for all nonexceptional affine types these polynomials satisfy a macdonald type duality

on new multiplicity formulas of weights of representations for the classical groups this is a continuation of 6 in this paper we give the multiplicity formulas of weights in an irreducible representation of the classical groups so n c and sp 2n c c gl 2n c using only young diagrams roughly speaking for each series of classical groups the weight multiplicities can be expressed as a polynomial function in the ranks in this direction r c king and s p 0 plunkett have presented the formulas of this kind using multinomial coefficients in s also see 4 in this paper we present another explicit expression of these polynomial functions moreover we characterize the set of weights dominant integral weights in each irreducible representation only in terms of young diagrams and also give the degrees of these polynomial functions and their leading coefficients our formulas are based on the known result concerning the so called kostka coefficients in the case of gl n and are a natural generalization from the general linear groups to the other classical groups as in the case of gl n the dominant integral weights of sg 2n 1 and sp 2n c gl 2n can be identified with the young diagrams see sect 1 in case of sg 2n the situation is a little bit delicate and for this case see section 1 for each series of the classical groups starting from these identifications we are able to regard the multiplicities of the dominant integral weight q in the irreducible representation with the highest weight 1 as the function of the rank n for fixed young diagrams 2 and q here each series of the classical groups means one of the following

fusion products of kirillov reshetikhin modules and the x m conjecture abstract in this article we show in the a d e case that the fusion product of kirillov reshetikhin modules for a current algebra whose character is expressed in terms of fermionic forms can be constructed from one dimensional modules by using joseph functors as a consequence we obtain some identity between fermionic forms and demazure operators since the same identity is also known to hold for one dimensional sums of nonexceptional type we can show from these results the x m conjecture for type a n 1 and d n 1

a positive combinatorial formula for symplectic kostka foulkes polynomials i rows abstract we prove a conjecture of lecouvey which proposes a closed positive combinatorial formula for symplectic kostka foulkes polynomials in the case of rows of arbitrary weight to show this we construct a new algorithm for computing cocyclage in terms of which the conjecture is described our algorithm is free of local constraints which were the main obstacle in lecouvey s original construction in particular we show that our model is governed by the situation in type a this approach works for arbitrary weight and we expect it to lead to a proof of the conjecture in full generality

the degenerate analogue of ariki s categorification theorem we explain how to deduce the degenerate analogue of ariki s categorification theorem over the ground field mathbb c as an application of schur weyl duality for higher levels and the kazhdan lusztig conjecture in finite type a we also discuss some supplementary topics including young modules tensoring with sign tilting modules and ringel duality

scattering rules in soliton cellular automata associated with uq d 1 n crystal bn 1 by means of the crystal theory we study a class of automata associated with uq d 1 n crystal bn 1 they have a commuting family of time evolutions and solitons of length l are labeled by uq a 1 n 1 crystal b2 la the scattering rule of two solitons of lengths l1 and l2 l1 l2 including the phase shift is identified with the combinatorial r matrix for the uq a 1 n 1 crystal b2 l2a b2 l1a

grobner shirshov bases for irreducible sln 1 modules abstract we determine the grobner shirshov bases for finite dimensional irreducible representations of the special linear lie algebra sl n 1 and construct explicit monomial bases for these representations we also show that each of these monomial bases is in 1 1 correspondence with the set of semistandard young tableaux of a given shape

crystals of type d_n 1 and young walls we give a new realization of arbitrary level perfect crystals and arbitrary level irreducible highest weight crystals of type d n 1 in the language of young walls we refine the notions of splitting of blocks and slices that have appeared in our previous works and these play crucial roles in the construction of crystals the perfect crystals are realized as the set of equivalence classes of slices and the irreducible highest weight crystals are realized as the affine crystals consisting of reduced proper young walls which in turn are concatenations of slices

limits of weight spaces lusztig s q analogs and fiberings of adjoint orbits let g be a connected complex semisimple algebraic group and t a maximal torus inside a borel subgroup b with g t and b their lie algebras let v be a representation in the category a for g the t decomposition v met va of v into a direct sum of finite dimensional weight spaces is central in the representation theory of g in this paper we introduce on weight spaces a new structure the principal filtration je v where e is a principal nilpotent in g chosen to be compatible with t for example e can be the sum of the simple root vectors relative to t b this filtration is constructed in a very simple way by taking j v to be the space of vectors annihilated by the p 1 th power of e for p 0 our approach is motivated by kostant s fundamental work ki k2 on actions of the principal tds three dimensional subalgebra and coordinate rings of regular adjoint orbits in theorem 3 4 we give a new description in terms of the dimension jumps of the principal filtration of the weight space of lusztig s l q analog mm q of dominant weight multiplicity in a finite dimensional irreducible g representation v the polynomial m q was defined algebraically as an alternating sum over the weyl group through a q analog of kostant s weight multiplicity formula we prove that m q is equal to the jump polynomial

the am 1 automata related to crystals of symmetric tensors a soliton cellular automaton associated with crystals of symmetric tensor representations of the quantum affine algebra uq am 1 is introduced it is a crystal theoretic formulation of the generalized box ball system in which capacities of boxes and carriers are arbitrary and inhomogeneous scattering matrices of two solitons coincide with the combinatorial r matrices of uq am 1 1 a piecewise linear evolution equation of the automaton is identified with an ultradiscrete limit of the nonautonomous discrete kadomtsev petviashivili equation a class of n soliton solutions is obtained through the ultradiscretization of soliton solutions of the latter

identities for classical group characters of nearly rectangular shape abstract we derive several identities that feature irreducible characters of the general linear the symplectic the orthogonal and the special orthogonal groups all of the identities feature characters that are indexed by shapes that are nearly rectangular by which we mean that the shapes are rectangles except for one row or column that might be shorter than the others as applications we prove new results in plane partitions and tableaux enumeration including new refinements of the bender knuth and macmahon ex conjectures

cohomology of flag varieties and the brylinski kostant filtration let g be a reductive algebraic group over c and let n be a g module for any subspace m of n the brylinski kostant filtration on m is defined through the action of a principal nilpotent element in lieg this filtration is related to a q analog of weight multiplicity due to lusztig we generalize this filtration to other nilpotent elements and show that this generalized filtration is related to parabolic versions of lusztig s q analog of weight multiplicity along the way we also generalize results of broer on cohomology vanishing of bundles on cotangent bundles of partial flag varieties we conclude by computing some explicit examples

fusion products of kirillov reshetikhin modules and fermionic multiplicity formulas abstract we give a complete description of the graded multiplicity space which appears in the feigin loktev fusion product of graded kirillov reshetikhin modules for all simple lie algebras this construction is used to obtain an upper bound formula for the fusion coefficients in these cases the formula generalizes the case of g a r of our previous paper where the multiplicities are generalized kostka polynomials in the case of other lie algebras the formula is the fermionic side of the x m conjecture in the cases where the kirillov reshetikhin conjecture regarding the decomposition formula for tensor products of kr modules has been proven in its original restricted form our result provides a proof of the conjectures of feigin and loktev regarding the fusion product multiplicities

crystal graphs for representations of the q analogue of classical lie algebras the explicit form of the crystal graphs for the finite dimensional representations of the q analogue of the universal enveloping algebras of type a b c and d is given in terms of semi standard tableaux and its analogues