quantum field theory and hopf algebra cohomology we exhibit a hopf superalgebra structure of the algebra of field operators of quantum field theory qft with the normal product based on this we construct the operator product and the time ordered product as a twist deformation in the sense of drinfeld our approach yields formulae for perturbative products and expectation values that allow for a significant enhancement in computational efficiency as compared to traditional methods employing hopf algebra cohomology sheds new light on the structure of qft and allows the extension to interacting not necessarily perturbative qft we give a reconstruction theorem for time ordered products in the spirit of streater and wightman and recover the distinction between free and interacting theory from a property of the underlying cocycle we also demonstrate how non trivial vacua are described in our approach solving a problem in quantum chemistry

algebraic and combinatorial structures on pairs of twin binary trees abstract we give a new construction of a hopf algebra defined first by reading 2005 rea05 whose bases are indexed by objects belonging to the baxter combinatorial family i e baxter permutations pairs of twin binary trees etc our construction relies on the definition of the baxter monoid analog of the plactic monoid and the sylvester monoid and on a robinson schensted like correspondence and insertion algorithm indeed the baxter monoid leads to the definition of a lattice structure over pairs of twin binary trees and the definition of a hopf algebra the algebraic properties of this hopf algebra are studied and among other multiplicative bases are provided and freeness and self duality proved

matching rota baxter algebras matching dendriform algebras and matching pre lie algebras abstract we introduce the notion of a matching rota baxter algebra motivated by the recent work on multiple pre lie algebras arising from the study of algebraic renormalization of regularity structures 10 19 this notion is also related to iterated integrals with multiple kernels and solutions of the associative polarized yang baxter equation generalizing the natural connection of rota baxter algebras with dendriform algebras to matching rota baxter algebras we obtain the notion of matching dendriform algebras as in the classical case of one operation matching rota baxter algebras and matching dendriform algebras are related to matching pre lie algebras which coincide with the aforementioned multiple pre lie algebras more general notions and results on matching tridendriform algebras and matching postlie algebras are also obtained

cumulants free cumulants and half shuffles free cumulants were introduced as the proper analogue of classical cumulants in the theory of free probability there is a mix of similarities and differences when one considers the two families of cumulants whereas the combinatorics of classical cumulants is well expressed in terms of set partitions that of free cumulants is described and often introduced in terms of non crossing set partitions the formal series approach to classical and free cumulants also largely differs the purpose of this study is to put forward a different approach to these phenomena namely we show that cumulants whether classical or free can be understood in terms of the algebra and combinatorics underlying commutative as well as non commutative half shuffles and half unshuffles as a corollary cumulants and free cumulants can be characterized through linear fixed point equations we study the exponential solutions of these linear fixed point equations which display well the commutative respectively non commutative character of classical and free cumulants

twisted descent algebras and the solomon tits algebra abstract the notion of descent algebra of a bialgebra is lifted to the barratt joyal setting of twisted bialgebras the new twisted descent algebras share many properties with their classical counterparts for example there are twisted analogs of classical lie idempotents and of the peak algebra moreover the universal twisted descent algebra is equipped with two products and a coproduct and there is a fundamental rule linking all three this algebra is shown to be naturally related to the geometry of the coxeter complex of type a

hopf algebra of the planar binary trees is a sub hopf algebra there is a basis qn of soln such that the composite map k qn soln k sn has the following property its linear dual k sn k qn is induced by a set theoretic map sn qn in between sn and qn there is an intermediate set yn which is made of planar binary trees with n vertices for instance for n 3 the number of elements in s3 y3 and q3 is 6 5 and 4 respectively the projection from sn to qn is the composite of two maps sn w yn w qn article no ai981759

hopf co addition for free magma algebras and the non associative hausdorff series abstract generalizations of the series exp and log to noncommutative non associative and other types of algebras were considered by m lazard and recently by v drensky and l gerritzen there is a unique power series exp x in one non associative variable x such that exp x exp x exp 2 x exp 0 1 we call the unique series h h x y in two non associative variables satisfying exp h exp x exp y the non associative hausdorff series and we show that the homogeneous components h n of h are primitive elements with respect to the co addition for non associative variables we describe the space of primitive elements for the co addition in non associative variables using taylor expansion and a projector onto the algebra a 0 of constants for the partial derivations by a theorem of kurosh a 0 is a free algebra we describe a procedure to construct a free algebra basis consisting of primitive elements

a categorification of biclosed sets of strings abstract we consider a closure space known as biclosed sets of strings of a gentle algebra of finite representation type palu pilaud and plamondon proved that the collection of all biclosed sets of strings forms a lattice and moreover that this lattice is congruence uniform many interesting examples of finite congruence uniform lattices may be represented as the lattice of torsion classes of an associative algebra we introduce a generalization the lattice of torsion shadows and we prove that the lattice of biclosed sets of strings is isomorphic to a lattice of torsion shadows when every indecomposable module over the gentle algebra is a brick finite congruence uniform lattices admit an alternate partial order known as the core label order in many cases the core label order of a congruence uniform lattice is isomorphic to a lattice of wide subcategories of an associative algebra analogous to torsion shadows we introduce wide shadows and prove that the core label order of the lattice of biclosed sets is isomorphic to a lattice of wide shadows

intervals of balanced binary trees in the tamari lattice we show that the set of balanced binary trees is closed by interval in the tamari lattice we establish that the intervals t t where t and t are balanced binary trees are isomorphic as posets to a hypercube we introduce synchronous grammars that allow to generate tree like structures and obtain fixed point functional equations to enumerate these we also introduce imbalance tree patterns and show that they can be used to describe some sets of balanced binary trees that play a particular role in the tamari lattice finally we investigate other families of binary trees that are also closed by interval in the tamari lattice

shuffle compatible permutation statistics abstract since the early work of richard stanley it has been observed that several permutation statistics have a remarkable property with respect to shuffles of permutations we formalize this notion of a shuffle compatible permutation statistic and introduce the shuffle algebra of a shuffle compatible permutation statistic which encodes the distribution of the statistic over shuffles of permutations this paper develops a theory of shuffle compatibility for descent statistics statistics that depend only on the descent set and length which has close connections to the theory of p partitions quasisymmetric functions and noncommutative symmetric functions we use our framework to prove that many descent statistics are shuffle compatible and to give explicit descriptions of their shuffle algebras thus unifying past results of stanley gessel stembridge aguiar bergeron nyman and petersen

construction de certaines operades et bigebres associees aux polytopes de stasheff et hypercubes stasheff polytopes introduced by stasheff in his study of h spaces are linked to associativity the direct sum of their cellular complexes is the underlying complex of the a operad which describes homotopy associative algebras in particular there exists a quasi isomorphism a as here we define on the direct sum of their dual cellular complexes the structure of a differential graded operad this construction extends the dendriform operad of loday which corresponds to the vertices of the polytopes we also define the structure of a differential graded operad on the direct sum of the dual cellular complexes of the hypercubes we define a quasi isomorphism from as to each of these operads we also define non differential variants of the two preceding operads and a morphism from as to each of these operads we show that the free algebras have a coproduct which turns them into bialgebras les polytopes de stasheff introduits pour l etude des h espaces sont lies a l associativite la somme directe de leurs complexes cellulaires forme le complexe sous jacent a l operade a qui decrit les algebres associatives a homotopie pres en particulier il existe un quasi isomorphisme a as ici on munit la somme directe des duaux de leurs complexes cellulaires d une structure d operade differentielle graduee cette construction generalise l operade des algebres dendriformes de loday qui correspond aux sommets des polytopes on munit aussi la somme directe des duaux des complexes cellulaires des hypercubes d une structure d operade differentielle graduee on definit un quasi isomorphisme de as clans chacune de ces deux operades on construit egalement des variantes non differentielles des deux operades precedentes on definit un morphisme de as dans chacune de ces operades et on montre que les algebres libres sont munies d un coproduit coassociatif qui en fait des bigebres

peak quasisymmetric functions and eulerian enumeration abstract via duality of hopf algebras there is a direct association between peak quasisymmetric functions and enumeration of chains in eulerian posets we study this association explicitly showing that the notion of cd index long studied in the context of convex polytopes and eulerian posets arises as the dual basis to a natural basis of peak quasisymmetric functions introduced by stembridge thus eulerian posets having a nonnegative cd index for example face lattices of convex polytopes correspond to peak quasisymmetric functions having a nonnegative representation in terms of this basis we diagonalize the operator that associates the basis of descent sets for all quasisymmetric functions to that of peak sets for the algebra of peak functions and study the g polynomial for eulerian posets as an algebra homomorphism

hopf algebras of rooted forests cocyles and free rota baxter algebras the hopf algebra and the rota baxter algebra are the two algebraic structures underlying the algebraic approach of connes and kreimer to renormalization of perturbative quantum field theory in particular the hopf algebra of rooted trees serves as the baby model of feynman graphs in their approach and can be characterized by certain universal properties involving a hochschild 1 cocycle decorated rooted trees have also been applied to study feynman graphs we will continue the study of universal properties of various spaces of decorated rooted trees with such a 1 cocycle leading to the concept of a cocycle hopf algebra we further apply the universal properties to equip a free rota baxter algebra with the structure of a cocycle hopf algebra

structure of the loday ronco hopf algebra of trees abstract loday and ronco defined an interesting hopf algebra structure on the linear span of the set of planar binary trees they showed that the inclusion of the hopf algebra of non commutative symmetric functions in the malvenuto reutenauer hopf algebra of permutations factors through their hopf algebra of trees and these maps correspond to natural maps from the weak order on the symmetric group to the tamari order on planar binary trees to the boolean algebra we further study the structure of this hopf algebra of trees using a new basis for it we describe the product coproduct and antipode in terms of this basis and use these results to elucidate its hopf algebraic structure in the dual basis for the graded dual hopf algebra our formula for the coproduct gives an explicit isomorphism with a free associative algebra we also obtain a transparent proof of its isomorphism with the non commutative connes kreimer hopf algebra of foissy and show that this algebra is related to non commutative symmetric functions as the commutative connes kreimer hopf algebra is related to symmetric functions

skew young diagram method in spectral decomposition of integrable lattice models the spectral decomposition of the path space of the vertex model associated to the vector representation of the quantized affine algebra is studied we give a one to one correspondence between the spin configurations and the semistandard tableaux of skew young diagrams as a result we obtain a formula of the characters for the degeneracy of the spectrum in terms of skew schur functions we conjecture that our result describes the module contents of the yangian module structures of the level 1 integrable modules of the affine lie algebra an analogous result is obtained also for a vertex model associated to the quantized twisted affine algebra where characters appear for the degeneracy of the spectrum the relations to the spectrum of the haldane shastry and the polychronakos models are also discussed

plane posets special posets and permutations abstract we study the self dual hopf algebra h sp of special posets introduced by malvenuto and reutenauer and the hopf algebra morphism from h sp to the hopf algebra of free quasi symmetric functions fqsym given by linear extensions in particular we construct two hopf subalgebras both isomorphic to fqsym the first one is based on plane posets the second one on heap ordered forests an explicit isomorphism between these two hopf subalgebras is also defined with the help of two combinatorial transformations on special posets the restriction of the hopf pairing of h sp to these hopf subalgebras and others is also studied as well as certain isometries between them these problems are solved using duplicial and dendriform structures

hyperoctahedral chen calculus for effective hamiltonians abstract we tackle the problem of unraveling the algebraic structure of computations of effective hamiltonians this is an important subject in view of applications to chemistry solid state physics or quantum field theory we show among other things that the correct framework for these computations is provided by the hyperoctahedral group algebras we define several structures on these algebras and give various applications for example we show that the adiabatic evolution operator in the time dependent interaction representation of an effective hamiltonian can be written naturally as a picard type series and has a natural exponential expansion

fermionic solution of the andrews baxter forrester model ii proof of melzer s polynomial identities we compute the one dimensional configuration sums of the abf model using the fermionic technique introduced in part i of this paper combined with the results of andrews baxter and forrester we prove polynomial identities for finitizations of the virasoro characters chi b a r 1 r q as conjectured by melzer in the thermodynamic limit these identities reproduce rogers ramanujan type identities for the unitary minimal virasoro characters conjectured by the stony brook group we also present a list of additional virasoro character identities which follow from our proof of melzer s identities and application of bailey s lemma

enriched p partitions and peak algebras abstract we develop a more general view of stembridge s enriched p partitions and use this theory to outline the structure of peak algebras for the symmetric group and the hyperoctahedral group initially we focus on commutative peak algebras spanned by sums of permutations with the same number of peaks where we consider several variations on the definition of peak whereas stembridge s enriched p partitions are related to quasisymmetric functions the dual coalgebra of solomon s type a descent algebra our generalized enriched p partitions are related to type b quasisymmetric functions the dual coalgebra of solomon s type b descent algebra using these functions we move on to explore non commutative peak algebras spanned by sums of permutations with the same set of peaks while some of these algebras have been studied before our approach gives explicit structure constants with a combinatorial description

the bloch wave operator generalizations and applications part i the time independent case this is part 1 of a two part review on wave operator theory and methods the basic theory of the time independent wave operator is presented in terms of partitioned matrix theory for the benefit of general readers with a discussion of the links between the matrix and projection operator approaches the matrix approach is shown to lead to simple derivations of the wave operators and effective hamiltonians of lowdin bloch des cloizeaux and kato as well as to some associated variational forms the principal approach used throughout stresses the solution of the nonlinear equation for the reduced wave operator leading to the construction of the effective hamiltonians of bloch and of des cloizeaux several mathematical techniques which are useful in implementing this approach are explained some of them being relatively little known in the area of wave operator calculations the theoretical discussion is accompanied by several specimen numerical calculations which apply the described techniques to a selection of test matrices taken from the previous literature on wave operator methods the main emphasis throughout is on the use of numerical methods which use iterative or perturbation algorithms with simple pade approximant methods being found sufficient to deal with most of the cases of divergence which are encountered the use of damping factors and relaxation parameters is found to be effective in stabilizing calculations which use the energy dependent effective hamiltonian of lowdin in general the computations suggest that the numerical applications of the nonlinear equation for the reduced wave operator are best carried out with the equation split into a pair of equations in which the bloch effective hamiltonian appears as a separate entity the presentation of the theoretical and computational details throughout is accompanied by references to and discussion of many works which have used wave operator methods in physics chemistry and engineering some of the techniques described in this part 1 will be further extended and applied in part 2 of the review which deals with the changes which are required to extend wave operator theory to the case of a time dependent hamiltonian such as that which describes the interaction of a laser pulse with an atom or molecule

triangulations orientals and skew monoidal categories abstract a concrete model of the free skew monoidal category fsk on a single generating object is obtained the situation is clubbable in the sense of g m kelly so this allows a description of the free skew monoidal category on any category as the objects of fsk are meaningfully bracketed words in the skew unit i and the generating object x it is necessary to examine bracketings and to find the appropriate kinds of morphisms between them this leads us to relationships between triangulations of polygons the tamari lattice left and right bracketing functions and the orientals a consequence of our description of fsk is a coherence theorem asserting the existence of a strictly structure preserving faithful functor fsk where is the skew monoidal category of finite non empty ordinals and first element and order preserving functions this in turn provides a complete solution to the word problem for skew monoidal categories

weak bruhat order on the set of faces of the permutohedron and the associahedron abstract we define a partial order on the set of faces of the permutohedron which extends the weak bruhat order of the symmetric group the restriction of this order gives a partially ordered set structure to the set of faces of the associahedron which extends the tamari order defined on the set of planar binary rooted trees these orders are used to describe associative algebra structures on the vector spaces spanned by the set of faces of the permutohedra on one hand and on the vector space spanned by the set of planar rooted trees on the other hand our results extend the description of the dendriform algebra structure on the spaces spanned by permutations and by planar binary trees in terms of the weak bruhat order and the tamari order respectively obtained by j l loday and the second author in j l loday m ronco order structure and the algebra of permutations and of planar binary trees j algebraic combin 15 2002 253 270

non commutative hopf algebra of formal diffeomorphisms abstract this paper deals with two hopf algebras which are the non commutative analogues of two different groups of formal power series the first group is the set of invertible series with the group law being multiplication of series while the second is the set of formal diffeomorphisms with the group law being a composition of series the motivation to introduce these hopf algebras comes from the study of formal series with non commutative coefficients invertible series with non commutative coefficients still form a group and we interpret the corresponding new non commutative hopf algebra as an alternative to the natural hopf algebra given by the co ordinate ring of the group which has the advantage of being functorial in the algebra of coefficients for the formal diffeomorphisms with non commutative coefficients this interpretation fails because in this case the composition is not associative anymore however we show that for the dual non commutative algebra there exists a natural co associative co product defining a non commutative hopf algebra moreover we give an explicit formula for the antipode which represents a non commutative version of the lagrange inversion formula and we show that its coefficients are related to planar binary trees then we extend these results to the semi direct co product of the previous hopf algebras and to series in several variables finally we show how the non commutative hopf algebras of formal series are related to some renormalization hopf algebras which are combinatorial hopf algebras motivated by the renormalization procedure in quantum field theory and to the renormalization functor given by the double tensor algebra on a bi algebra

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

a solomon descent theory for the wreath products _ we propose an analogue of solomon s descent theory for the case of a wreath product g g n where g is a finite abelian group our construction mixes a number of ingredients mantaci reutenauer algebras specht s theory for the representations of wreath products okada s extension to wreath products of the robinson schensted correspondence and poirier s quasisymmetric functions we insist on the functorial aspect of our definitions and explain the relation of our results with previous work concerning the hyperoctaedral group

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

a method for proving multiterminal source coding theorems the two step method used by wyner and ziv to prove the wyner ziv theorem is extended to prove sliding block source coding theorems for coding a general finite alphabet ergodic multiterminal source with respect to a single letter fidelity criterion the first step replaces every stochastic encoding in the network by a deterministic sliding block encoding the second step involves using the slepian wolf theorem to adjust the sliding block encodings so that they will have the desired rate while introducing little additional distortion the method is applied to give quick proofs of sliding block versions of theorems of berger kaspi and tung since the method obtains sliding block coders directly without first obtaining block coders the block coding results can be obtained as an easy corollary the direct methods for proving results on block coding are more difficult and do not imply the corresponding results on sliding block coding indeed in the multiterminal case it is in general not known how to construct good sliding block coders from good block coders

random walks on quasisymmetric functions abstract conditions are provided under which an endomorphism on quasisymmetric functions gives rise to a left random walk on the descent algebra which is also a lumping of a left random walk on permutations spectral results are also obtained several important random walks are now realized this way stanley s qs distribution results from endomorphisms given by evaluation maps a shuffles result from the a th convolution power of the universal character and the tchebyshev operator of the second kind introduced recently by ehrenborg and readdy yields traditional riffle shuffles a conjecture of ehrenborg regarding the spectra for a family of random walks on ab words is proven a theorem of stembridge from the theory of enriched p partitions is also recovered as a special case

combinatorics of cyclic shifts in plactic hypoplactic sylvester baxter and related monoids abstract the cyclic shift graph of a monoid is the graph whose vertices are elements of the monoid and whose edges link elements that differ by a cyclic shift this paper examines the cyclic shift graphs of plactic like monoids whose elements can be viewed as combinatorial objects of some type aside from the plactic monoid itself the monoid of young tableaux examples include the hypoplactic monoid quasi ribbon tableaux the sylvester monoid binary search trees the stalactic monoid stalactic tableaux the taiga monoid binary search trees with multiplicities and the baxter monoid pairs of twin binary search trees it was already known that for many of these monoids connected components of the cyclic shift graph consist of elements that have the same evaluation that is contain the same number of each generating symbol this paper focuses on the maximum diameter of a connected component of the cyclic shift graph of these monoids in the rank n case for the hypoplactic monoid this is n 1 for the sylvester and taiga monoids at least n 1 and at most n for the stalactic monoid 3 except for ranks 1 and 2 when it is respectively 0 and 1 for the plactic monoid at least n 1 and at most 2 n 3 the current state of knowledge including new and previously known results is summarized in a table

a decomposition of the descent algebra of the hyperoctahedral group elements of the hyperoctahedral group bn can be represented as lists of integers 1 2 n where the list of absolute value of the k s is a permutation of the integers from 1 to n the descent set of such a is des k 0 n 1 k k 1 where k 0 we set 0 0 a descent class as of bn is the formal sum of all elements of bn with a given descent set s 0 1 n 1 in this paper we shall give a combinatorial proof of a result of solomon 6 stating that the product in the group algebra q bn of two descent classes is a linear combination of descent classes we refer to the linear span of these descent classes it is a subalgebra bn of q bn as solomon s hyperoctahedral descent algebra we derive from our combinatorial multiplication rule for bn that there is a family of ideals jv n of bn such that j0 n j1 n jn n with bn v 1 bnjv n in 4 garsia and reutenauer relate a similar decomposition of the multiplicative structure of the descent algebra of the symmetric group sn to the action of the symmetric group on the free lie algebra it develops that one can introduce a bn analog of the free lie algebra and give similar results for bn

a multiplication rule for the descent algebra of typed here we give an interpretation of solomon s rule for multiplication in the descent algebra of coxeter groups of typed dn we describe an ideal i such that dn i is isomorphic to the descent algebra of the hyperoctahedral group bn 2

a set operad of formal fractions and dendriform like sub operads abstract we introduce an operad of formal fractions abstracted from the mould operads and containing both the dendriform and the tridendriform operads we consider the smallest set operad contained in this operad and containing four specific elements of arity two corresponding to the generators and the associative elements of the dendriform and tridendriform operads we obtain a presentation of this operad by binary generators and quadratic relations and an explicit combinatorial description using a new kind of bi coloured trees similar results are also presented for related symmetric operads

the normal distribution is infinitely divisible abstract we prove that the classical normal distribution is infinitely divisible with respect to the free additive convolution we study the voiculescu transform first by giving a survey of its combinatorial implications and then analytically including a proof of free infinite divisibility in fact we prove that a sub family of askey wimp kerov distributions are freely infinitely divisible of which the normal distribution is a special case at the time of this writing this is only the third example known to us of a nontrivial distribution that is infinitely divisible with respect to both classical and free convolution the others being the cauchy distribution and the free 1 2 stable distribution

the peak algebra and the descent algebras of types b and d we show the existence of a unital subalgebra n of the symmetric group algebra linearly spanned by sums of permutations with a common peak set which we call the peak algebra we show that n is the image of the descent algebra of type b under the map to the descent algebra of type a which forgets the signs and also the image of the descent algebra of type d the algebra n contains a two sided ideal n which is defined in terms of interior peaks this object was introduced in previous work by nyman 2003 we find that it is the image of certain ideals of the descent algebras of types b and d we derive an exact sequence of the form 0 n n n 2 0 we obtain this and many other properties of the peak algebra and its peak ideal by first establishing analogous results for signed permutations and then forgetting the signs in particular we construct two new commutative semisimple subalgebras of the descent algebra of dimensions n and n 2 1 by grouping permutations according to their number of peaks or interior peaks we discuss the hopf algebraic structures that exist on the direct sums of the spaces n and n over n 0 and explain the connection with previous work of stembridge 1997 we also obtain new properties of his descents to peaks map and construct a type b analog

a decomposition of solomon s descent algebra abstract a descent class in the symmetric group sn is the collection of permutations with a given descent set it was shown by l solomon j algebra41 1976 255 264 that the product in the group algebra q sn of two descent classes is a linear combination of descent classes thus descent classes generate a subalgebra of q sn we refer to it here as solomon s descent algebra and denote it by n this algebra is not semisimple but it has a faithul representation in terms of upper triangular matrices the main goal of this paper is a decomposition of its multiplicative structure it develops that n acts in a natural way on the so called lie monomials this action has a purely combinatorial description and is a crucial tool in the construction of a complete set of indecomposable representations of n in particular we obtain a natural basis of irreducible orthogonal idempotents indexed by partitions of n for the quotient n n natural bases of nilpotents and idempotents for the subspaces e ne for two arbitrary partitions and are also constructed and the dimensions of these spaces are given a combinatorial interpretation in terms of the so called decreasing factorization of an arbitary word into a product of lyndon words

the algebra of binary search trees we introduce a monoid structure on the set of binary search trees by a process very similar to the construction of the plactic monoid the robinson schensted insertion being replaced by the binary search tree insertion this leads to a new construction of the algebra of planar binary trees of loday ronco defining it in the same way as non commutative symmetric functions and free symmetric functions we briefly explain how the main known properties of the loday ronco algebra can be described and proved with this combinatorial point of view and then discuss it from a representation theoretical point of view which in turns leads to new combinatorial properties of binary trees

groups of tree expanded series abstract in ch brouder a frabetti renormalization of qed with planar binary trees eur phys j c 19 2001 715 741 ch brouder a frabetti qed hopf algebras on planar binary trees j algebra 267 2003 298 322 we introduced three hopf algebras on planar binary trees related to the renormalization of quantum electrodynamics one of them the algebra h is commutative and is therefore the ring of coordinate functions of a proalgebraic group g the other two algebras h e and h are free non commutative therefore their abelian quotients are the coordinate rings of two proalgebraic groups g e and g in this paper we describe explicitly these groups using two monoidal structures and a set operad structure on planar binary trees we show that these groups can be realized on formal series expanded over trees and that the group laws are generalizations of the multiplication and the composition of usual series in one variable therefore we obtain some new groups of invertible tree expanded series and of tree expanded formal diffeomorphisms respectively the hopf algebra describing the renormalization of the electric charge corresponds to the subgroup of tree expanded formal diffeomorphisms formed of the translations which fix the zero by some particular tree expanded series which remind the proper correlation functions in quantum field theory in turn the group of tree expanded formal diffeomorphisms and some of its subgroups give rise to new hopf algebras on trees all the constructions are done in a general operad theoretic setting and then applied to the specific duplicial operad on trees

the shifted poirier reutenauer algebra based on the shifted schensted correspondence and the shifted knuth equivalence a shifted analog of the poirier reutenauer algebra is introduced it is a right coideal subalgebra of the poirier reutenauer algebra and turns out to be a higher lift of schur s p functions its close relations with the peak subalgebra and the stembridge algebra of peak functions are also uncovered

jordan trialgebras and post jordan algebras abstract we compute minimal sets of generators for the s n modules n 4 of multilinear polynomial identities of arity n satisfied by the jordan product and the jordan diproduct resp pre jordan product in every triassociative resp tridendriform algebra these identities define jordan trialgebras and post jordan algebras jordan analogues of the lie trialgebras and post lie algebras introduced by dotsenko et al pei et al vallette loday we include an extensive review of analogous structures existing in the literature and their interrelations in order to identify the gaps filled by our two new varieties of algebras we use computer algebra linear algebra over finite fields representation theory of symmetric groups to verify in both cases that every polynomial identity of arity 6 is a consequence of those of arity 4 we conjecture that in both cases the next independent identities have arity 8 imitating the glennie identities for jordan algebras we formulate our results as a commutative square of operad morphisms which leads to the conjecture that the squares in a much more general class are also commutative

characters for from a novel thermodynamic bethe ansatz motivated by the recent development on the exact thermodynamics of one dimensional quantum systems we propose quasiparticle like formulae for characters the case is re examined first the novel formulation yields a direct connection to the fractional statistics in the short range interacting model and provides a clear description of the spinon character formula generalizing the observation we find formulae for which can be proved by the durfee rectangle formula

skew schubert functions and the pieri formula for flag manifolds we show the equivalence of the pieri formula for flag manifolds with certain identities among the structure constants for the schubert basis of the polynomial ring this gives new proofs of both the pieri formula and of these identities a key step is the association of a symmetric function to a finite poset with labeled hasse diagram satisfying a symmetry condition this gives a unified definition of skew schur functions stanley symmetric functions and skew schubert functions defined here we also use algebraic geometry to show the coefficient of a monomial in a schubert polynomial counts certain chains in the bruhat order obtainng a combinatorial chain construction of schubert polynomials

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

hidden yangians in 2d massive current algebras we define non local conserved currents in massive current algebras in two dimensions our approach is algebraic and non perturbative the non local currents give a quantum field realization of the yangians we show how the noncocommutativity of the yangians is related to the non locality of the currents we discuss the implications of the existence of non local conserved charges on thes matrices

spectral decomposition of path space in solvable lattice model we give thespectral decomposition of the path space of the u q widehat sl 2 vertex model with respect to the local energy functions the result suggests the hidden yangian module structure on the widehat sl 2 levell integrable modules which is consistent with the earlier work 1 in the level one case also we prove the fermionic character formula of the widehat sl 2 levell integrable representations in consequence

free tri dendriform family algebras abstract in this paper we first prove that a rota baxter family algebra indexed by a semigroup amounts to an ordinary rota baxter algebra structure on the tensor product with the semigroup algebra we show that the same phenomenon arises for dendriform and tridendriform family algebras then we construct free dendriform family algebras in terms of typed decorated planar binary trees finally we generalize typed decorated rooted trees to typed valently decorated schroder trees and use them to construct free tridendriform family algebras

the peak algebra of the symmetric group revisited abstract the linear span p n of the sums of all permutations in the symmetric group sn with a given set of peaks is a sub algebra of the symmetric group algebra due to nyman this peak algebra is a left ideal of the descent algebra d n and the direct sum p of all p n is a hopf sub algebra of the direct sum d of all d n dual to the stembridge algebra of peak functions in our self contained approach peak counterparts of several results on the descent algebra are established including a simple combinatorial characterization of the algebra p n an algebraic characterization of p n based on the action on the poincare birkhoff witt basis of the free associative algebra the display of peak variants of the classical lie idempotents an eulerian type sub algebra of p n a description of the jacobson radical of p n and its nil potency index of the principal indecomposable and irreducible p n modules and of the cartan matrix of p n furthermore it is shown that the primitive lie algebra of p is free and that p is its enveloping algebra

qed hopf algebras on planar binary trees abstract in this paper we describe the hopf algebras on planar binary trees used to renormalize the feynman propagators of quantum electrodynamics and the coaction which describes the renormalization procedure both structures are related to some semi direct coproduct of hopf algebras

algebres de hopf des permutahedres associahedres et hypercubes resume three graded hopf algebras over z are constructed each one having in degree n a basis indexed by the faces of a polytope of dimension n 1 respectively the hypercube the associahedron and the permutahedron this lifts to the level of all faces previous constructions of loday and ronco and malvenuto and reutenauer for vertices of the same polyhedra

coalgebras of words and phrases abstract we introduce two constructions of a coassociative comultiplication in the algebra of phrases in a given alphabet as a preliminary step we give two constructions of a pre lie comultiplication in the module generated by words

on hopf algebra structures over free operads the operad lie can be constructed as the operad of primitives primas from the operad as of associative algebras this is reflected by the theorems of friedrichs poincare birkhoff witt and cartier milnor moore we replace the operad as by families of free operads p which include the operad mag freely generated by a non commutative non associative binary operation and the operad of stasheff polytopes we obtain poincare birkhoff witt type theorems and collect information about the operads primp e g in terms of characteristic functions

tree expansion in time dependent perturbation theory the computational complexity of time dependent perturbation theory is well known to be largely combinatorial whatever the chosen expansion method and family of parameters combinatorial sequences goldstone and other feynman type diagrams etc we show that a very efficient perturbative expansion both for theoretical and numerical purposes can be obtained through an original parametrization by trees and generalized iterated integrals we emphasize above all the simplicity and naturality of the new approach that links perturbation theory with classical and recent results in enumerative and algebraic combinatorics these tools are applied to the adiabatic approximation and the effective hamiltonian we prove perturbatively and nonperturbatively the convergence of morita s generalization of the gell mann and low wave function we show that summing all the terms associated with the same tree leads to an utter simplification where the sum is simpler than any of its terms finally we recover the rayleigh schrodinger time independent equation for the wave operator and we give an explicit nonrecursive expression for the term corresponding to an arbitrary tree

schubert polynomials for the classical groups the task of a theory of schubert polynomials is to produce explicit representatives for schubert classes in the cohomology ring of a flag variety and to do so in a manner that is as natural as possible from a combinatorial point of view to explain more fully let us review a special case the schubert calculus for grassmannians where one asks for the number of linear spaces of given dimension satisfying certain geometric conditions a typical problem is to find the number of lines meeting four given lines in general position in 3 space answer below for each of the four given lines the set of lines meeting it is a schubert variety in the grassmannian and we want the number of intersection points of these four subvarieties in the modem solution of this problem the schubert varieties induce canonical elements of the cohomology ring of the grassmannian called schubert classes the product of these schubert classes is the class of a point times the number of intersection points counted with appropriate multiplicities this reformulation of the problem though one of the great achievements of algebraic geometry is only part of a solution it remains to give a concrete model for the cohomology ring that makes explicit computation with schubert classes possible as it happens the cohomology rings of grassmannians can be identified with quotients of a polynomial ring so that schubert classes correspond to schur functions intersection numbers such as we are considering then turn out to be littlewood richardson coefficients for example the answer to our four lines 4 problem is the coefficient of the schur function s 2 2 in the product s 1 x or 2 for an extended treatment and history of the subject see 10 11 18 the identification of schur functions as schubert polynomials for grassmannians is a consequence of a more general and now highly developed theory of schubert polynomials for the flag varieties of the special linear groups sl n c the starting point for this more general theory is a construction of

cambrian hopf algebras abstract cambrian trees are oriented and labeled trees which fulfill local conditions around each node generalizing the classical conditions for binary search trees similar to binary trees for the tamari lattice cambrian trees provide convenient combinatorial models for n reading s cambrian lattices of type a based on a natural surjection from signed permutations to cambrian trees we define the cambrian hopf algebra extending j l loday and m ronco s algebra on binary trees we describe combinatorially the products and coproducts of both the cambrian algebra and its dual in terms of operations on cambrian trees we then construct the baxter cambrian algebra which extends s law and n reading s baxter hopf algebra on rectangulations and s giraudo s equivalent hopf algebra on twin binary trees

poisson geometrical symmetries associated to non commutative formal diffeomorphisms let open image in new window be the group of all formal power series starting with x with coefficients in a field open image in new window of zero characteristic with the composition product and let f open image in new window be its function algebra in bf a non commutative non cocommutative graded hopf algebra open image in new window was introduced via a direct process of disabelianisation of f open image in new window taking the like presentation of the latter as an algebra but dropping the commutativity constraint in this paper we apply a general method to provide four one parameter deformations of open image in new window which are quantum groups whose semiclassical limits are poisson geometrical symmetries such as poisson groups or lie bialgebras namely two quantum function algebras and two quantum universal enveloping algebras in particular the two poisson groups are extensions of open image in new window isomorphic as proalgebraic poisson varieties but not as proalgebraic groups

tridendriform structure on combinatorial hopf algebras we extend the definition of tridendriform bialgebra by introducing a parameter q the subspace of primitive elements of a q tridendriform bialgebra is equipped with an associative product and a natural structure of brace algebra related by a distributive law this data is called q gerstenhaber voronov algebras we prove the equivalence between the categories of conilpotent q tridendriform bialgebras and of q gerstenhaber voronov algebras the space spanned by surjective maps between finite sets as well as the space spanned by parking functions have a natural structure of q tridendriform bialgebra denoted st q and pqsym q in such a way that st q is a sub tridendriform bialgebra of pqsym q finally we show that the bialgebra of m permutations defined by t lam and p pylyavskyy comes from a q tridendriform algebra which is a quotient of st q