modules of the 0 hecke algebra and quasisymmetric schur functions abstract we begin by deriving an action of the 0 hecke algebra on standard reverse composition tableaux and use it to discover 0 hecke modules whose quasisymmetric characteristics are the natural refinements of schur functions known as quasisymmetric schur functions furthermore we classify combinatorially which of these 0 hecke modules are indecomposable from here we establish that the natural equivalence relation arising from our 0 hecke action has equivalence classes that are isomorphic to subintervals of the weak bruhat order on the symmetric group focussing on the equivalence classes containing a canonical tableau we discover a new basis for the hopf algebra of quasisymmetric functions and use the cardinality of these equivalence classes to establish new enumerative results on truncated shifted reverse tableau studied by panova and adin king roichman generalizing our 0 hecke action to one on skew standard reverse composition tableaux we derive 0 hecke modules whose quasisymmetric characteristics are the skew quasisymmetric schur functions of bessenrodt et al this enables us to prove a restriction rule that reflects the coproduct formula for quasisymmetric schur functions which in turn yields a quasisymmetric branching rule analogous to the classical branching rule for schur functions

weak quasi symmetric functions rota baxter algebras and hopf algebras abstract we introduce the hopf algebra of quasi symmetric functions with semigroup exponents generalizing the hopf algebra qsym of quasi symmetric functions as a special case we obtain the hopf algebra qsym n of weak quasi symmetric functions which provides a framework for the study of a question proposed by g c rota relating symmetric type functions and rota baxter algebras we provide the transformation formulas between the weak monomial and fundamental quasi symmetric functions which extends the corresponding results for quasi symmetric functions moreover we show that qsym is a hopf subalgebra and a hopf quotient algebra of qsym n rota s question is addressed by identifying qsym n with the free commutative unitary rota baxter algebra x of weight 1 on one generator x which also allows us to equip x with a hopf algebra structure

analysis of the objective space in multiple objective linear programming abstract a multiple objective linear program is defined by a matrix c consisting of the coefficients of the linear objectives and a convex polytope x defined by the linear constraints an analysis of the objective space y c x for this problem is presented a characterization between a face of y and the corresponding faces of x is obtained this result gives a necessary and sufficient condition for a face to be efficient the theory and examples demonstrate the collapsing simplification that occurs in mapping x to y these results form a basis for a new approach to analyzing multiple objective linear programs

a unification of knizhnik zamolodchikov and dunkl operators via affine hecke algebras some generalizations of the lusztig lascoux schutzenberger operators for affine hecke algebras are considered as corollaries we obtain lusztig s isomorphisms from affine hecke algebras to their degenerate versions a natural interpretation of the dunkl operators and a new class of differential difference operators generalizing dunkl s ones and the knizhnik zamolodchikov operators from the two dimensional conformal field theory

finding all maximal efficient faces in multiobjective linear programming an algorithm for finding the whole efficient set of a multiobjective linear program is proposed from the set of efficient edges incident to a vertex a characterization of maximal efficient faces containing the vertex is given by means of the lexicographic selection rule of dantzig orden and wolfe a connectedness property of the set of dual optimal bases associated to a degenerate vertex is proved an application of this to the problem of enumerating all the efficient edges incident to a degenerate vertex is proposed our method is illustrated with numerical examples and comparisons with armand malivert s algorithm show that this new algorithm uses less computer time

nica toeplitz algebras associated with product systems over right lcm semigroups abstract we prove uniqueness of representations of nica toeplitz algebras associated to product systems of c correspondences over right lcm semigroups by applying our previous abstract uniqueness results developed for c precategories our results provide an interpretation of conditions identified in work of fowler and fowler raeburn and apply also to their crossed product twisted by a product system in the new context of right lcm semigroups as well as to a new doplicher roberts type c algebra associated to the nica toeplitz algebra as a derived construction we develop nica toeplitz crossed products by actions with completely positive maps this provides a unified framework for nica toeplitz semigroup crossed products by endomorphisms and by transfer operators we illustrate these two classes of examples with semigroup c algebras of right and left semidirect products

discrete product systems and twisted crossed products by semigroups a product systemeover a semigrouppis a family of hilbert spaces es s p together with multiplicationses et est we vieweas a unitary valued cocycle onp and consider twisted crossed productsa e pinvolvingeand an action ofpby endomorphisms of ac algebraa whenpis quasi lattice ordered in the sense of nica we isolate a class of covariant representations ofe and consider a twisted crossed productbp e pwhich is universal for covariant representations ofewhenehas finite dimensional fibres and in general is slightly larger in particular whenp nand dim e1 our algebrabn e n is a new infinite analogue of the toeplitz cuntz algebras ton our main theorem is a characterisation of the faithful representations ofbp e p

product systems over right angled artin semigroups we build upon mac lane s definition of a tensor category to introduce the concept of a product system that takes values in a tensor groupoid g we show that the existing notions of product systems fit into our categorical framework as do the k graphs of kumjian and pask we then specialize to product systems over right angled artin semigroups these are semigroups that interpolate between free semigroups and free abelian semigroups for such a semigroup we characterize all product systems which take values in a given tensor groupoid g in particular we obtain necessary and sufficient conditions under which a collection of k 1 graphs form the coordinate graphs of a k graph

semigroups of endomorphisms dirichlet series and phase transitions abstract we study phase transitions of c dynamical systems a in which a is the crossed product of a c algebra by a lattice semigroup of endomorphisms and is a one parameter subgroup of the dual action determined by a real valued scale on the semigroup we show that the kms equilibrium condition is equivalent to a markov type condition on the predual semigroup dynamical system and give criteria for the existence of equilibrium states and of phase transitions in terms of the semigroup action when the semigroup is an integer lattice the partition function of the real scale is a dirichlet series having an euler product expansion on their convergence region there is a phase transition whose geometry is fully determined by the ground states and is independent of the chosen scale generalizing a model recently constructed by bost and connes motivated by earlier work of julia as applications we simplify part of the proof the bost connes theorem on phase transition with spontaneous symmetry breaking and we discuss generalizations to other systems associated to subsets of primes and to number fields of class number 1

hecke algebras difference operators and quasi symmetric functions abstract we define a new action of the symmetric group and its hecke algebra on polynomial rings whose invariants are exactly the quasi symmetric polynomials we interpret this construction in terms of a demazure character formula for the irreducible polynomial modules of a degenerate quantum group we use the action of the generic hecke algebras to define quasi symmetric and noncommutative analogues of hall littlewood functions we show that these generalized functions share many combinatorial properties with the classical ones nous introduisons de nouvelles actions du groupe symetrique et de son algebre de hecke sur les polynomes pour lesquelles les invariants sont les polynomes quasi symetriques nous interpretons cette construction en termes de caracteres de demazure d un groupe quantique degenere nous utilisons l action de l algebre de hecke generique pour definir des analogues quasi symetriques et non commutatifs des fonctions de hall littlewood nous montrons que ces fonctions generalisees ont un certain nombre de proprietes communes avec les fonctions classiques

skew quasisymmetric schur functions and noncommutative schur functions abstract recently a new basis for the hopf algebra of quasisymmetric functions qsym called quasisymmetric schur functions has been introduced by haglund luoto mason van willigenburg in this paper we extend the definition of quasisymmetric schur functions to introduce skew quasisymmetric schur functions these functions include both classical skew schur functions and quasisymmetric schur functions as examples and give rise to a new poset l c that is analogous to young s lattice we also introduce a new basis for the hopf algebra of noncommutative symmetric functions nsym this basis of nsym is dual to the basis of quasisymmetric schur functions and its elements are the pre image of the schur functions under the forgetful map nsym sym we prove that the multiplicative structure constants of the noncommutative schur functions equivalently the coefficients of the skew quasisymmetric schur functions when expanded in the quasisymmetric schur basis are nonnegative integers satisfying a littlewood richardson rule analogue that reduces to the classical littlewood richardson rule under as an application we show that the morphism of algebras from the algebra of poirier reutenauer to sym factors through nsym we also extend the definition of schur functions in noncommuting variables of rosas sagan in the algebra ncsym to define quasisymmetric schur functions in the algebra ncqsym we prove these latter functions refine the former and their properties and project onto quasisymmetric schur functions under the forgetful map lastly we show that by suitably labeling l c skew quasisymmetric schur functions arise in the theory of pieri operators on posets

c envelopes of semicrossed products by lattice ordered abelian semigroups abstract a semicrossed product is a non selfadjoint operator algebra encoding the action of a semigroup on an operator or c algebra we prove that when the positive cone of a discrete lattice ordered abelian group acts on a c algebra the c envelope of the associated semicrossed product is a full corner of a crossed product by the whole group by constructing a c cover that itself is a full corner of a crossed product and computing the shilov ideal we obtain an explicit description of the c envelope this generalizes a result of davidson fuller and kakariadis from z n to the class of all discrete lattice ordered abelian groups

ideals of quasi symmetric functions and super covariant polynomials for sn the aim of this work is to study the quotient ring r n of the ring q x 1 x n over the ideal j n generated by non constant homogeneous quasi symmetric functions we prove here that the dimension of r n is given by c n the n th catalan number this is also the dimension of the space sh n of super covariant polynomials that is defined as the orthogonal complement of j n with respect to a given scalar product we construct a basis for r n whose elements are naturally indexed by dyck paths this allows us to understand the hilbert series of sh n in terms of number of dyck paths with a given number of factors

semigroup crossed products and the toeplitz algebras of nonabelian groups abstract we consider the quasi lattice ordered groups g p recently introduced by nica we realise their universal toeplitz algebra as a crossed product b p p by a semigroup of endomorphisms and show that the toeplitz representation is faithful precisely when an amenability condition is satisfied we then show directly that many interesting free products are amenable the final result contains theorems of cuntz and dinh on the uniqueness of toeplitz cuntz algebras as special cases

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

optimization over the efficient set of a parametric multiple objective linear programming problem abstract in this paper we consider the problem p for optimizing a function over the efficient set of a multiple objective linear programming molp problem with parameters in the right hand side vector three solution algorithms in a most general case of problem p and improvements of them in some special cases are presented in the case when the objective function of problem p is linear it can be solved based on t 2 linear programming problems with mixed one zero integer variables if the parametric set is finite and based on t 2 linear programming problems if the right hand side vector of the molp problem is a linear function of the parameters and the parametric set is a polyhedron where t 2 is in general the number of maximal efficient faces of the molp problem corresponding to a value of the parameters a numerical example is given to illustrate the working of the algorithms

combinatorial hopf algebras noncommutative hall littlewood functions and permutation tableaux abstract we introduce a new family of noncommutative analogues of the hall littlewood symmetric functions our construction relies upon tevlin s bases and simple q deformations of the classical combinatorial hopf algebras we connect our new hall littlewood functions to permutation tableaux and also give an exact formula for the q enumeration of permutation tableaux of a fixed shape this gives an explicit formula for the steady state probability of each state in the partially asymmetric exclusion process pasep the polynomial enumerating permutations with a fixed set of weak excedances according to crossings the polynomial enumerating permutations with a fixed set of descent bottoms according to occurrences of the generalized pattern 2 31

tensor algebras of product systems and their c envelopes abstract let g p be an abelian lattice ordered group and let x be a compactly aligned product system over p with coefficients in a we show that the c envelope of the nica tensor algebra n t x coincides with both sehnem s covariance algebra a x p and the co universal c algebra n o x r for injective gauge compatible nica covariant representations of carlsen larsen sims and vittadello we give several applications of this result on both the selfadjoint and non selfadjoint operator algebra theory first we guarantee the existence of n o x r thus settling a problem of carlsen larsen sims and vittadello which was open even for abelian lattice ordered groups as a second application we resolve a problem posed by skalski and zacharias on dilating isometric representations of product systems to unitary representations as a third application we characterize the c envelope of the tensor algebra of a finitely aligned higher rank graph which also holds for topological higher rank graphs as a final application we prove reduced hao ng isomorphisms for generalized gauge actions of discrete groups on c algebras of product systems this generalizes recent results that were obtained by various authors in the case where g p z n

representations of product systems over semigroups and dilations of commuting cp maps we prove that every pair of commuting cp maps on a von neumann algebra m can be dilated to a commuting pair of endomorphisms on a larger von neumann algebra to achieve this we first prove that every completely contractive representation of a product system of c correspondences over the semigroup n2 can be dilated to an isometric or toeplitz representation

a graded representation model for macdonald s polynomials we define doubly graded sn modules rmu for which we conjecture that the multiplicities of irreducible representations in various bi degrees are given by the macdonald coefficients klambdamu assuming one fundamental conjecture the modules rmu can be given several equivalent definitions which we discuss we prove the conjectures in various special cases

duality between quasi symmetrical functions and the solomon descent algebra abstract the ring qsym of quasi symmetric functions is naturally the dual of the solomon descent algebra the product and the two coproducts of the first extending those of the symmetric functions correspond to a coproduct and two products of the second which are defined by restriction from the symmetric group algebra a consequence is that qsym is a free commutative algebra

determining maximal efficient faces in multiobjective linear programming problem finding an efficient or weakly efficient solution in a multiobjective linear programming molp problem is not a difficult task the difficulty lies in finding all these solutions and representing their structures since there are many convenient approaches that obtain all of the weakly efficient extreme points and weakly efficient extreme rays in an molp this paper develops an algorithm which effectively finds all of the weakly efficient maximal faces in an molp using all of the weakly efficient extreme points and extreme rays the proposed algorithm avoids the degeneration problem which is the major problem of the most of previous algorithms and gives an explicit structure for maximal efficient weak efficient faces consequently it gives a convenient representation of efficient weak efficient set using maximal efficient weak efficient faces the proposed algorithm is based on two facts firstly the efficiency and weak efficiency property of a face is determined using a relative interior point of it secondly the relative interior point is achieved using some affine independent points indeed the affine independent property enable us to obtain an efficient relative interior point rapidly

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

applications of the wold decomposition to the study of row contractions associated with directed graphs based on a wold decomposition for families of partial isometries and projections of cuntz krieger toeplitz type we extend several fundamental theorems from the case of single vertex graphs to the general case of countable directed graphs with no sinks we prove a szego type factorization theorem for ckt families which leads to information on the structure of the unit ball in free semigroupoid algebras and show that joint similarity implies joint unitary equivalence for such families for each graph we prove a generalization of von neumann s inequality which applies to row contractions of operators on hilbert space which are related to the graph in a natural way this yields a functional calculus determined by quiver algebras and free semigroupoid algebras we establish a generalization of coburn s theorem for the c algebra of a ckt family and prove a universality theorem for c algebras generated by these families in both cases the c algebras generated by quiver algebras play the universal role

constructing efficient solutions structure of multiobjective linear programming it is not a difficult task to find a weak pareto or pareto solution in a multiobjective linear programming molp problem the difficulty lies in finding all these solutions and representing their structure this paper develops an algorithm for solving this problem we investigate the solutions and their relationships in the objective space the algorithm determines finite number of weights each of which corresponds to a weighted sum problems by solving these problems we further obtain all weak pareto and pareto solutions of the molp and their structure in the constraint space the algorithm avoids the degeneration problem which is a major hurdle of previous works and presents an easy and clear solution structure

the maximal descriptor index set for a face of a convex polyhedral set and some applications abstract a face of a convex polyhedral set can be investigated on the basis of only the maximal descriptor index set for it this helps us reduce difficulties and computational efforts in solving many problems in this paper we consider a multiple objective linear programming molp problem in which the constraint polyhedron may have no extreme points and the variables may be negative first we deal with determining the maximal descriptor index set for an arbitrary face of the constraint polyhedron then we apply the maximal descriptor index sets for faces to compare descriptor sets for the faces and find all maximal efficient faces for the molp problem the methods known so far compare descriptor sets at large computational cost in terms of computing times and storage because they require finding all extreme points and directions of them known top down search methods have been devised to find the efficient set of the molp problem but they may fail to find all maximal efficient faces and the computational cost is considerable we propose a new method to compare descriptor sets and a new top down search method which finds all maximal efficient faces and requires less computational cost a numerical example is given to illustrate how the method works

topological freeness for c correspondences abstract we study conditions that ensure uniqueness theorems of cuntz krieger type for relative cuntz pimsner algebras o j x associated to a c correspondence x over a c algebra a we give general sufficient conditions phrased in terms of a multivalued map x acting on the spectrum a of a when x j is of type i we construct a directed graph dual to x and prove a uniqueness theorem using this graph when x j is liminal we show that topological freeness of this graph is equivalent to the uniqueness property for o j x as well as to an algebraic condition which we call j acyclicity of x as an application we improve the fowler raeburn uniqueness theorem for the toeplitz algebra t x we give new simplicity criteria for o x we generalize and enhance uniqueness results for relative quiver c algebras of muhly and tomforde we also discuss applications to crossed products by endomorphisms

the deformed virasoro algebra at roots of unity we discuss some aspects of the representation theory of the deformed virasoro algebra virpq in particular we give a proof of the formula for the kac determinant and then determine the center of virp q for q a primitive nth root of unity we derive explicit expressions for the generators of the center in the limit t qp 1 and elucidate the connection to the hall littlewood symmetric functions furthermore we argue that for the algebra describes gentile statistics of order n 1 i e a situation in which at most n 1 particles can occupy the same state

discrete product systems and their c algebras continuous product systems or product systems over r were recently introduced by arveson to develope an index theory for continuous semigroups of endomorphisms of b h in this paper we continue his work by studying discrete product systems or product systems over g where g is a countable dense subgroup of r we give examples of discrete product systems and compute their automorphism groups to each discrete product system e over g we associate a separable c algebra oe g whose non degenerate representations correspond bijectively to representations of e this family of c algebras may be regarded as analogues of cuntz algebras or car algebras in some degenerate cases these c algebras are the generalized toeplitz algebras we show that oe g is simple and study its gauge automorphisms quasi free automorphisms and fock state