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We study the ring of quasisymmetric polynomials in $n$ anticommuting (fermionic) variables. Let $R_n$ denote the polynomials in $n$ anticommuting variables. The main results of this paper show the following interesting facts about quasisymmetric polynomials in anticommuting variables: (1) The quasisymmetric polynomials in $R_n$ form a commutative sub-algebra of $R_n$. (2) There is a basis of the quotient of $R_n$ by the ideal $I_n$ generated by the quasisymmetric polynomials in $R_n$ that is indexed by ballot sequences. The Hilbert series of the quotient is given by $$ \text{Hilb}_{R_n/I_n}(q) = \sum_{k=0}^{\lfloor{n/2}\rfloor} f^{(n-k,k)} q^k\,,$$ where $f^{(n-k,k)}$ is the number of standard tableaux of shape $(n-k,k)$. (3) There is a basis of the ideal generated by quasisymmetric polynomials that is indexed by sequences that break the ballot condition, Comment: 17 pages, minor corrections to paper, including remarks from Darij Grinberg

We construct three new combinatorial Hopf algebras based on the Loday-Ronco operations on planar binary trees. The first and second algebras are defined on planar trees and labeled planar trees extending the Loday-Ronco and Malvenuto-Reutenauer Hopf algebras respectively. We show that the latter is bidendriform which implies that is also free, cofree, and self-dual. The third algebra involves a new visualization of parking functions as decorated binary trees; it is also bidendriform, free, cofree, and self-dual, and therefore abstractly isomorphic to the algebra PQSym of Novelli and Thibon. We define partial orders on the objects indexing each of these three Hopf algebras, one of which, when restricting to (m+1)-ary trees, coarsens the m-Tamari order of Bergeron and Pr\'eville-Ratelle. We show that multiplication of dual fundamental basis elements are given by intervals in each of these orders. Finally, we use an axiomatized version of the techniques of Aguiar and Sottile on the Malvenuto-Reutenauer Hopf algebra to define a monomial basis on each of our Hopf algebras, and to show that comultiplication is cofree on the monomial elements. This in particular, implies the cofreeness of the Hopf algebra on planar trees. We also find explicit positive formulas for the multiplication on monomial basis and a cancellation-free and grouping-free formula for the antipode of monomial elements.

In an earlier paper, the first two authors defined orientations on hypergraphs. Using this definition we provide an explicit bijection between acyclic orientations in hypergraphs and faces of hypergraphic polytopes. This allows us to obtain a geometric interpretation of the coefficients of the antipode map in a Hopf algebra of hypergraphs. This interpretation differs from similar ones for a different Hopf structure on hypergraphs provided recently by Aguiar and Ardila. Furthermore, making use of the tools and definitions developed here regarding orientations of hypergraphs we provide a characterization of hypergraphs giving rise to simple hypergraphic polytopes in terms of acyclic orientations of the hypergraph. In particular, we recover this fact for the nestohedra and the hyper-permutahedra, and prove it for generalized Pitman-Stanley polytopes as defined here., v2 contains minor revisions and matches published version

We introduce a Hopf algebra structure of subword complexes, including both finite and infinite types. We present an explicit cancellation free formula for the antipode using acyclic orientations of certain graphs, and show that this Hopf algebra induces a natural non-trivial sub-Hopf algebra on c-clusters in the theory of cluster algebras.

For any $n > 0$ and $0 \leq m < n$, let $P_{n,m}$ be the poset of projective equivalence classes of $\{-,0,+\}$-vectors of length $n$ with sign variation bounded by $m$, ordered by reverse inclusion of the positions of zeros. Let $\Delta_{n,m}$ be the order complex of $P_{n,m}$. A previous result from the third author shows that $\Delta_{n,m}$ is Cohen-Macaulay over $\mathbb{Q}$ whenever $m$ is even or $m = n-1$. Hence, it follows that the $h$-vector of $\Delta_{n,m}$ consists of nonnegative entries. Our main result states that $\Delta_{n,m}$ is partitionable and we give an interpretation of the $h$-vector when $m$ is even or $m = n-1$. When $m = n-1$ the entries of the $h$-vector turn out to be the new Eulerian numbers of type $D$ studied by Borowiec and Młotkowski in [ Electron. J. Combin., 23(1):#P1.38, 2016]. We then combine our main result with Klee's generalized Dehn-Sommerville relations to give a geometric proof of some facts about these Eulerian numbers of type $D$.

In 2009, Sagan and Savage introduced a combinatorial model for the Fibonomial numbers, integer numbers that are obtained from the binomial coefficients by replacing each term by its corresponding Fibonacci number. In this paper, we present a combinatorial description for the $q$-analog and elliptic analog of the Fibonomial numbers. This is achieved by introducing some $q$-weights and elliptic weights to a slight modification of the combinatorial model of Sagan and Savage.

La journaliste Louise Alphonsine Nantel (1884-1965) a redige de nombreuses chroniques dans L'Avenir du Nord, Le Canada, Le Pays et La Patrie signees Andree Claudel, Mireille, Lou Sorriaux ou L. A. Nantel. Le present article se penche sur ses premiers ecrits, de 1904 a 1914. Nee a Saint-Jerome, Louise Alphonsine demenage a Montreal a 19 ans puis s'installe a Paris en 1911, d'ou elle poursuit sa collaboration aux journaux montrealais. On lui doit des portraits saisissants de la vie urbaine, et, si elle publie bon nombre de chroniques qu'on peut qualifier de mondaines, elle demeure critique envers la frivolite et l'hypocrisie de la bourgeoisie. Femme moderne et emancipee, elle appartient au mouvement progressiste; elle soutient les revendications feministes ainsi que la participation des femmes a la vie publique et plaide pour une bibliotheque publique et l'appui du gouvernement a l'education. Elle epouse le commissaire de police parisien Julien Sorriaux. En 1930, le couple s'etablit a Saint-Tropez, ou la journaliste meurt en 1966. En 1947, elle publie un roman autobiographique sur son enfance dans les Laurentides. Abstract: Journalist Louise Alphonsine Nantel (1884–1965) published many columns in L'Avenir du Nord, Le Canada, Le Pays and La Patrie under the names of Andree Claudel, Mireille, Lou Sorriaux and L.A. Nantel. This article examines her early work (1904–1914). Born in Saint-Jerome, Louise Alphonsine moved to Montreal at the age of 19 and then settled in Paris in 1911, from where she continued collaborating with Montreal newspapers. She is the author of vivid portraits of urban life, and while she published a large number of "social columns," she remained critical of the bourgeoisie's frivolity and hypocrisy. A modern and emancipated woman, she belonged to the progressive movement: she supported feminist demands as well as the participation of women in public life, and advocated for a public library and the government's support of education. She married a Parisian chief of police Julien Sorriaux. In 1930, the couple moved to Saint-Tropez where the journalist passed away in 1966. In 1947, she published an autobiographical novel about her childhood in the Laurentides.

We introduce the theory of ( P , ρ ) -partitions, depending on a poset P and a map ρ from P to positive integers. The generating function F P , ρ of ( P , ρ ) -partitions is a polynomial that, when the images of ρ tend to infinity, tends to Stanley’s generating function of P -partitions. Analogous to Stanley’s fundamental theorem for P -partitions, we show the set of ( P , ρ ) -partitions decomposes as a disjoint union of ( L , ρ ) -partitions where L runs over the set of linear extensions of P . In this more general context, the set of all F L , ρ for linear orders L over determines a basis of polynomials. We thus introduce the notion of flagged ( P , ρ ) -partitions, and we prove the set of all F L , ρ for flagged ( L , ρ ) -partitions for linear orders L is precisely the fundamental slide basis of the polynomial ring, introduced by the first author and Searles. Our main theorem shows that any generating function F P , ρ of flagged ( P , ρ ) -partitions is a positive integer linear combination g slide polynomials. As applications, we give a new proof of positivity of the slide product and, motivating our nomenclature, we also prove flagged Schur functions are slide positive.

We study the symmetric function and polynomial combinatorial invariants of Hopf algebras of permutations, posets and graphs. We investigate their properties and the relations among them. In particular, we show that the chromatic symmetric function and many other invariants have a property we call positively $h$-alternating. This property of positively $h$-alternating leads to Schur positivity and $e$-positivity when applying the operator $\nabla$ at $q=1$. We conclude by showing that the invariants we consider can be expressed as scheduling problems., Comment: 26 pages, some colors, first final draft comments welcome

We present complete simplicial fan realizations of any spherical subword complex of type $A_n$ for $n\leq 3$. This provides complete simplicial fan realizations of simplicial multi-associahedra $\Delta_{2k+4,k}$, whose facets are in correspondence with $k$-triangulations of a convex $(2k+4)$-gon. This solves the first open case of the problem of finding fan realizations where polytopality is not known. The techniques presented in this paper work for all finite Coxeter groups and we hope that they will be useful to construct fans realizing subword complexes in general. In particular, we present fan realizations of two previously unknown cases of subword complexes of type $A_4$, namely the multi-associahedra $\Delta_{9,2}$ and $\Delta_{11,3}$.

This paper introduces a Hopf algebra structure on a family of reduced pipe dreams. We show that this Hopf algebra is free and cofree, and construct a surjection onto a commutative Hopf algebra of permutations. The pipe dream Hopf algebra contains Hopf subalgebras with interesting sets of generators and Hilbert series related to subsequences of Catalan numbers. Three other relevant Hopf subalgebras include the Loday-Ronco Hopf algebra on complete binary trees, a Hopf algebra related to a special family of lattice walks on the quarter plane, and a Hopf algebra on $\nu$-trees related to $\nu$-Tamari lattices. One of this Hopf subalgebras motivates a new notion of Hopf chains in the Tamari lattice, which are used to present applications and conjectures in the theory of multivariate diagonal harmonics., Comment: 44 pages, 29 figures; Version 4: minor changes for final version

We introduce a new basis of the non-commutative symmetric functions whose commutative images are Schur functions. Dually, we build a basis of the quasi-symmetric functions which expand positively in the fundamental quasi-symmetric functions and decompose Schur functions. We then use the basis to construct a non-commutative lift of the Hall-Littlewood symmetric functions with similar properties to their commutative counterparts., Comment: new version includes edits, references to forthcoming research, and a 'hook-length' formula for the number of standard immaculate tableaux

We give an $AB$-factorization of the supercharacter table of the group of $n\times n$ unipotent upper triangular matrices over $\FF_q$, where $A$ is a lower-triangular matrix with entries in $\ZZ[q]$ and $B$ is a unipotent upper-triangular matrix with entries in $\ZZ[q^{-1}]$. To this end we introduce a $q$ deformation of a new power-sum basis of the Hopf algebra of symmetric functions in noncommutative variables. The factorization is obtain from the transition matrices between the supercharacter basis, the $q$-power-sum basis and the superclass basis. This is similar to the decomposition of the character table of the symmetric group $S_n$ given by the transition matrices between Schur functions, monomials and power-sums. We deduce some combinatorial results associated to this decomposition. In particular we compute the determinant of the supercharacter table., (example added) 10 pages, Final version To Appear in int. J. of Algebra and Computation (Accepted 2012)

We give several explicit combinatorial formulas for the expansion of k-Schur functions indexed by maximal rectangles in terms of the standard basis of the affine nilCoxeter algebra. Using our result, we also show a commutation relation of k-Schur functions corresponding to rectangles with the generators of the affine nilCoxeter algebra., Comment: to appear in Journal of Combinatorics, 28 pages

The operator @? of F. Bergeron, Garsia, Haiman and Tesler [F. Bergeron, A. Garsia, M. Haiman, G. Tesler, Identities and positivity conjectures for some remarkable operators in the theory of symmetric functions, Methods Appl. Anal. 6 (1999) 363-420] acting on the k-Schur functions [L. Lapointe, A. Lascoux, J. Morse, Tableaux atoms and a new Macdonald positivity conjecture, Duke Math. J. 116 (2003) 103-146; L. Lapointe, J. Morse, Schur functions analogs for a filtration of the symmetric functions space, J. Combin. Theory Ser. A 101 (2003) 191-224; L. Lapointe, J. Morse, Tableaux on k+1-cores, reduced words for affine permutations and k-Schur expansion, J. Combin. Theory Ser. A 112 (2005) 44-81] indexed by a single column has a coefficient in the expansion which is an analogue of the (q,t)-Catalan number with a level k. When k divides n we conjecture a representation theoretical model in this case such that the graded dimensions of the module are the coefficients of the (q,t)-Catalan polynomials of level k. When the parameter t is set to 1, the Catalan numbers of level k are shown to count the number of Dyck paths that lie below a certain Dyck path with q counting the area of the path.

The Grothendieck group of the tower of symmetric group algebras has a self-dual graded Hopf algebra structure. Inspired by this, we introduce by way of axioms, a general notion of a tower of algebras and study two Grothendieck groups on this tower linked by a natural paring. Using representation theory, we show that our axioms give a structure of graded Hopf algebras on each Grothendieck groups and these structures are dual to each other. We give some examples to indicate why these axioms are necessary. We also give auxiliary results that are helpful to verify the axioms. We conclude with some remarks on generalized towers of algebras leading to a structure of generalized bialgebras (in the sense of Loday) on their Grothendieck groups.

A combinatorial Hopf algebra is a graded connected Hopf algebra over a field $F$ equipped with a character (multiplicative linear functional) $\zeta:H\to F$. We show that the terminal object in the category of combinatorial Hopf algebras is the algebra $QSym$ of quasi-symmetric functions; this explains the ubiquity of quasi-symmetric functions as generating functions in combinatorics. We illustrate this with several examples. We prove that every character decomposes uniquely as a product of an even character and an odd character. Correspondingly, every combinatorial Hopf algebra $(H,\zeta)$ possesses two canonical Hopf subalgebras on which the character $\zeta$ is even (respectively, odd). The odd subalgebra is defined by certain canonical relations which we call the generalized Dehn-Sommerville relations. We show that, for $H=QSym$, the generalized Dehn-Sommerville relations are the Bayer-Billera relations and the odd subalgebra is the peak Hopf algebra of Stembridge. We prove that $QSym$ is the product (in the categorical sense) of its even and odd Hopf subalgebras. We also calculate the odd subalgebras of various related combinatorial Hopf algebras: the Malvenuto-Reutenauer Hopf algebra of permutations, the Loday-Ronco Hopf algebra of planar binary trees, the Hopf algebras of symmetric functions and of non-commutative symmetric functions., Comment: 34 pages

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., Comment: LaTeX, 3 figures, 12 pages

We introduce a Hopf algebra structure of subword complexes, including both finite and infinite types. We present an explicit cancellation free formula for the antipode using acyclic orientations of certain graphs, and show that this Hopf algebra induces a natural non-trivial sub-Hopf algebra on $c$-clusters in the theory of cluster algebras., Comment: Several color figures, 31 pages

We present complete simplicial fan realizations of any spherical subword complex of type $A_n$ for $n\leq 3$. This provides complete simplicial fan realizations of simplicial multi-associahedra $\Delta_{2k+4,k}$, whose facets are in correspondence with $k$-triangulations of a convex $(2k+4)$-gon. This solves the first open case of the problem of finding fan realizations where polytopality is not known. The techniques presented in this paper work for all finite Coxeter groups and we hope that they will be useful to construct fans realizing subword complexes in general. In particular, we present fan realizations of two previously unknown cases of subword complexes of type $A_4$, namely the multi-associahedra $\Delta_{9,2}$ and $\Delta_{11,3}$., Nous construisons des éventails simpliciaux complets ayant la combinatoire des complexes de sous-mots de type $A_n$ pour $n\leq 3$. Par conséquent, nous obtenons des constructions d’éventails des multi-associaèdres $\Delta_{2k+4,k}$, dont les facettes correspondent aux $k$-triangulations d’un $(2k+4)$-gone. Cette construction confirme l’existence d’éventails ayant la combinatoire du multi-associaèdres pour une famille dont la polytopalité n’est pas confirmée. Les techniques utilisées fonctionnent pour tous les groupes de Coxeter et nous espérons qu’elles seront utiles afin de construire des éventails réalisant les complexes de sous-mots en général. En particulier, nous présentons des éventails pour deux complexes de sous-mots de type $A_4$ dont l’existence était inconnue: les multi-associaèdres $\Delta_{9,2}$ et $\Delta_{11,3}$.

We investigate the quotient ring $R$ of the ring of formal power series $\Q[[x_1,x_2,...]]$ over the closure of the ideal generated by non-constant quasi-\break symmetric functions. We show that a Hilbert basis of the quotient is naturally indexed by Catalan paths (infinite Dyck paths). We also give a filtration of ideals related to Catalan paths from $(0,0)$ and above the line $y=x-k$. We investigate as well the quotient ring $R_n$ of polynomial ring in $n$ variables over the ideal generated by non-constant quasi-symmetric polynomials. We show that the dimension of $R_n$ is bounded above by the $n$th Catalan number.

We give the formula for multiplying a Schubert class on an odd orthogonal or symplectic flag manifold by a special Schubert class pulled back from a Grassmannian of maximal isotropic subspaces. This is also the formula for multiplying a type $B$ (respectively, type $C$) Schubert polynomial by the Schur $P$-polynomial $p_m$ (respectively, the Schur $Q$-polynomial $q_m$). Geometric constructions and intermediate results allow us to ultimately deduce this from formulas for the classical flag manifold. These intermediate results are concerned with the Bruhat order of the Coxeter group ${\mathcal B}_\infty$, identities of the structure constants for the Schubert basis of cohomology, and intersections of Schubert varieties. We show these identities follow from the Pieri-type formula, except some `hidden symmetries' of the structure constants. Our analysis leads to a new partial order on the Coxeter group ${\mathcal B}_\infty$ and formulas for many of these structure constants.

The main purpose of this paper is to show that the multiplication of a Schubert polynomial of finite type $A$ by a Schur function, which we refer to as Schubert vs. Schur problem, can be understood from the multiplication in the space of dual $k$-Schur functions. Using earlier work by the second author, we encode both problems by means of quasisymmetric functions. On the Schubert vs. Schur side, we study the poset given by the Bergeron-Sottile's $r$-Bruhat order, along with certain operators associated to this order. On the other side, we connect this poset with a graph on dual $k$-Schur functions given by studying the affine grassmannian order of Lam-Lapointe-Morse-Shimozono. Also, we define operators associated to the graph on dual $k$-Schur functions which are analogous to the ones given for the Schubert vs. Schur problem., 21 figure, some color in picture

Given a finite graded poset with labeled Hasse diagram, we construct a quasi- symmetric generating function for (saturated) chains whose labels have fixed descents. This is a common generalization of a generating function for the flag f-vector defined by Ehrenborg and of a symmetric function associated to certain edge-labeled posets which arose in the theory of Schubert polynomials. We show this construction gives a Hopf morphism from an incidence algebra of edge-labeled posets to the Hopf algebra of quasi-symmetric functions., Comment: LaTeX-2e, 9 pages. J. Algebra, to appear. Added examples, corrected typos, revised contents

We explore monoids generated by operators on certain infinite partial orders. Our starting point is the work of Fomin and Greene on monoids satisfying the relations \((\mathbf{u}_{r} + \mathbf{u}_{r+1})\mathbf{u}_{r+1}\mathbf{u}_{r} = \mathbf{u}_{r+1}\mathbf{u}_{r}(\mathbf{u}_{r} + \mathbf{u}_{r+1})\) and \(\mathbf{u}_{r}\mathbf{u}_{t} = \mathbf{u}_{t}\mathbf{u}_{r}\) if | r − t | > 1. Given such a monoid, the non-commutative functions in the variables u are shown to commute. Symmetric functions in these operators often encode interesting structure constants. Our aim is to introduce similar results for more general monoids not satisfying the relations of Fomin and Greene. This paper is an extension of a talk by the second author at the workshop on algebraic monoids, group embeddings and algebraic combinatorics at The Fields Institute in 2012.

The immaculate basis of the non-commutative symmetric functions was recently introduced by the first and third author to lift certain structures in the symmetric functions to the dual Hopf algebras of the non-commutative and quasi-symmetric functions. It was shown that immaculate basis satisfies a positive, multiplicity free right Pieri rule. It was conjectured that the left Pieri rule may contain signs but that it would be multiplicity free. Similarly, it was also conjectured that the dual quasi-symmetric basis would also satisfy a signed multiplicity free Pieri rule. We prove these two conjectures here., 14 pages, corrections from v1 and v2

We continue our development of a new basis for the algebra of non-commutative symmetric functions. This basis is analogous to the Schur basis for the algebra of symmetric functions, and it shares many of its wonderful properties. For instance, in this article we describe non-commutative versions of the Littlewood-Richardson rule and the Murnaghan-Nakayama rule. A surprising relation develops among non-commutative Littlewood-Richardson coefficients, which has implications to the commutative case. Finally, we interpret these new coefficients geometrically as the number of integer points inside a certain polytope., 30 pages: we cleaned and fixed many details in the proofs. The interested reader may toggle \specialcomments in the TeX file to reveal 6 added pages of details and ideas (in red)

The main purpose of this paper is to show that the multiplication of a Schubert polynomial of finite type $A$ by a Schur function can be understood from the multiplication in the space of dual $k$-Schur functions. Using earlier work by the second author, we encode both problems by means of quasisymmetric functions. On the Schubert vs. Schur side, we study the $r$-Bruhat order given by Bergeron-Sottile, along with certain operators associated to this order. On the other side, we connect this poset with a graph on dual $k$-Schur functions given by studying the affine grassmannian order of Lam-Lapointe-Morse-Shimozono. Also, we define operators associated to the graph on dual $k$-Schur functions which are analogous to the ones given for the Schubert vs. Schur problem., Le but principal de cet article est de montrer que la multiplication d’un polynôme de Schubert de type fini $A$ par une fonction de Schur peut être comprise à partir de la multiplication dans l’espace dual des fonctions $k$-Schur. Les travaux antérieurs par le second auteur, nous permet de coder ces deux problèmes au moyen de fonctions quasi-symétriques. Du côté Schubert vs Schur, nous étudions l’ordre partiel $r$-Bruhat donné par Bergeron-Sottile, ainsi que certains opérateurs associés à cet ordre. Nous donnons une relation entre l’ordre $r$-Bruhat et le graphe de Bruhat sur les fonctions $k$-Schur duales données par l’étude de l’ordre affine grassmannienne de Lam-Lapointe-Morse-Shimozono. En outre, nous définissons des opérateurs associés a ce graphe qui sont analogues à ceux donnés pour le problème Schubert vs Schur.

From a recent paper, we recall the Hopf monoid structure on the supercharacters of the unipotent uppertriangular groups over a finite field. We give cancelation free formula for the antipode applied to the bases of class functions and power sum functions, giving new cancelation free formulae for the standard Hopf algebra of supercharacters and symmetric functions in noncommuting variables. We also give partial results for the antipode on the supercharacter basis, and explicitly describe the primitives of this Hopf monoid.

The variational method in a reformulated Hamiltonian formalism of Quantum Electrodynamics is used to derive relativistic wave equations for systems consisting of n fermions and antifermions of various masses. The derived interaction kernels of these equations include one-photon exchange interactions. The equations have the expected Schroedinger non-relativistic limit. Application to some exotic few lepton systems is discussed briefly., Comment: 22 pages

Let A be a commutative algebra over a field of characteristic zero, and M be a symmetric A-bimodule. Gerstenhaber and Schack have shown that there are Hodge-type decompositions Hn(A, M) = ⊕kHk, n − k(A, M), Hn(A, M) = ⊕kHk, n − k(A, M) of the Hochschild (co)homology. The first summands H1, n − 1(A, M), H1, n − 1(A, M) are known to be the Harrison (co)homology defined in terms of shuffles. We discuss interpretations of the decompositions in terms of k-shuffles and how these relate to versions of the Poincaré-Birkoff-Witt theorem. We then turn to a detailed study of how the decomposition behaves with respect to the Gerstenhaber operations (cup and Lie products) in cohomology. We show by example that neither product is generally graded, but that Fq = ⊕r ≥ q H∗,r(A, A) are ideals for both products with Fp ∪Fq⊆Fp + q and [Fp, Fq]⊆Fp + q. The results for the cup product were conjectured by Gerstenhaber and Schack.

Nous etudions des operateurs du groupe hyperoctahedral sur l'homologie de Hochschild. Plus precisement, nous etudions les familles d'operateurs de la categorie hyperoctahedrale qui commutent avec les operateurs de bord de Hochschild b n = ∑ i = 0 n (−1) i d i ou di: [n] → [n-1] est l'application de face dans cette categorie. Cette etude est intimement reliee aux projections dans les espaces de puissances symetriques et hyperoctahedrales d'algebres de Lie libres. Nous discuterons aussi de l'homologie de Harrison dans un tel contexte et de certains autres problemes connexes.

We construct Families {ρ l, k n } of orthogonal idempotents of the hyperoctahedral group algebras Q [ B n ], which commute with the Hochschild boundary operators b n =∑ n i=0 (−1) i d i . We show that those idempotents are projections onto some hyperoactahedral symmetric powers of the free Lie algebra Lie ( l, k ) n ( A ). The commutations above then decompose the Hochshild homology H n ( C ) obtained by any functor C : Δ op → K - module that factor through Fin ′ B , the hyperoctahedral category. Moreover, we show that this decomposition is the finest possible for any such functor. In particular, the Hochschild homology of a commutative algebra equipped with an involutive automorphism splits into components indexed by ( l, k ) and the corresponding Harrison homology splits into two components indexed by (0, 1) and (1, 0). Generalizing the Harrison complex, we show that H ( l,k ) n ( C )≅ H n (Sh ( l,k ) . /Sh ( l −1, k +1) . , where Sh ( r,s ) . are some shuffle complexes associated to C . We also give the characters of the representations related to Lie ( l, k ) n ( A ) as a direct sum of induced characters.

We build, from the collection of all groups of unitriangular matrices, Hopf monoids in Joyal's category of species. Such structure is carried by the collection of class function spaces on those groups, and also by the collection of superclass function spaces, in the sense of Diaconis and Isaacs. Superclasses of unitriangular matrices admit a simple description from which we deduce a combinatorial model for the Hopf monoid of superclass functions, in terms of the Hadamard product of the Hopf monoids of linear orders and of set partitions. This implies a recent result relating the Hopf algebra of superclass functions on unitriangular matrices to symmetric functions in noncommuting variables. We determine the algebraic structure of the Hopf monoid: it is a free monoid in species, with the canonical Hopf structure. As an application, we derive certain estimates on the number of conjugacy classes of unitriangular matrices., Final Version, 32 pages, accepted in "Algebra and Number Theory"

We construct a recursive formula for a complete system of primitive orthogonal idempotents for any R-trivial monoid. This uses the newly proved equivalence between the notions of R-trivial monoid and weakly ordered monoid., Nous construisons une formule récursive pour un système complet d'idempotents orthogonaux primitifs pour tout monoïde R-trivial. Nous employons une nouvelle équivalence entre les notions de monoïde R-trivial et de monoïde faiblement ordonné.

We identify two seemingly disparate structures: supercharacters, a useful way of doing Fourier analysis on the group of unipotent uppertriangular matrices with coefficients in a finite field, and the ring of symmetric functions in noncommuting variables. Each is a Hopf algebra and the two are isomorphic as such. This allows developments in each to be transferred. The identification suggests a rich class of examples for the emerging field of combinatorial Hopf algebras., Nous montrons que deux structures en apparence bien différentes peuvent être identifiées: les super-caractères, qui sont un outil commode pour faire de l'analyse de Fourier sur le groupe des matrices unipotentes triangulaires supérieures à coefficients dans un corps fini, et l'anneau des fonctions symétriques en variables non-commutatives. Ces deux structures sont des algèbres de Hopf isomorphes. Cette identification permet de traduire dans une structure les dévelopements conçus pour l'autre, et suggère de nombreux exemples dans le domaine nouveau des algèbres de Hopf combinatoires.

We give a type free formula for the expansion of $k$-Schur functions indexed by fundamental coweights within the affine nilCoxeter algebra. Explicit combinatorics are developed in affine type $C$.

Using a formula of Billey, Jockusch and Stanley, Fomin and Kirillov have introduced a new set of diagrams that encode the Schubert polynomials. We call these objects rc-graphs. We define and prove two variants of an algorithm for constructing the set of all rc-graphs for a given permutation. This construction makes many of the identities known for Schubert polynomials more apparent, and yields new ones. In particular, we give a new proof of Monk's rule using an insertion algorithm on rc-graphs. We conjecture two analogs of Pieri's rule for multiplying Schubert polynomials. We also extend the algorithm to generate the double Schubert polynomials.

In this paper we expose how algebraic explorations with a symbolic manipulation package (Maple) led the authors (after the work of Garsia and Reutenauer, 1989 ) to the construction of nice basis for the descent algebras of finite Coxeter groups. This resulted in a better understanding of the multiplicative structure of many of these descent algebras. We also expose how this work was used to work out the proofs of theorems and set up the conjectures for the structure of these algebras.

We show a combinatorial rule based on diagrams (finite nonempty sets of lattice points (i, j) in the positive quadrant) for the construction of the Schubert polynomials. In the particular case where the Schubert polynomial is a Schur function we give a bijection between our diagrams and column strict tableaux. A different algorithm had been conjectured (and proved in the case of vexillary permutations) by A. Kohnert (Ph.D. dissertation, Universitat auf Bayreuth, 1990). We give, at the end of this paper, a sketch of how one would show the equivalence of the two rules.

We begin by briefly recalling some of our previous results on the descent algebra of the hyperoctahedral groups Bn. From this we construct a ‘nice’ expression for the generating function of a family of orthogonal idempotents ϱkn. More precisely, ∑ k=1 n ρ k n x k = 1 2 n n! ∑ π∈B n (x−2d(π))↑ B n π , where d(π) stands for the number of descents of π and (x)↑ B n = (x + 1)(x + 3)(x + 5)⋯(x + 2n −1 ) . We show that the dimension of the right ideal Q [Bn]ϱkn is given by the number c(n,k) of elements of Bn having k positive cycles. Thus the dimension is given by an analogue of the Stirling numbers of the first kind. We also show that the algebra Λ generated by the classes of elements of Bn having equal numbers of descents is commutative. This relates to Loday's work on cyclic homology and Hochschild homology. In fact we show that we can define some λ operators on Λ which commute with the Hochschild boundary operator. This gives a decomposition for commutative hyperoctahedral algebra homology. We conclude our paper by presenting results about the hyperoctahedral shuffle algebra which are extensions of aspects of the work of Aldous, Bayer and Diaconis related to the shuffle algebra.

The purpose of this paper is twofold. First we aim to unify previous work by the first two authors, A. Garsia, and C. Reutenauer (see l2r, l3r, l4r, l5r and l10r) on the structure of the descent algebras of the Coxeter groups of type An and Bn. But we shall also extend these results to the descent algebra of an arbitrary finite Coxeter group W. The descent algebra, introduced by Solomon in l14r, is a subalgebra of the group algebra of W. It is closely related to the subring of the Burnside ring B(W) spanned by the permutation representations W/WJ, where the WJ are the parabolic subgroups of W. Specifically, our purpose is to lift a basis of primitive idempotents of the parabolic Burnside algebra to a basis of idempotents of the descent algebra.

We use the Hamiltonian formalism of quantum field theory and the variational basis-state method to derive relativistic three-body wave equations for scalar particles interacting via a massive or massless mediating scalar field (the scalar Yukawa model). A variational trial state comprised of three and five Fock-space states is used to derive coupled wave equations for a relativistic three- (and five-) body system. Approximate variational three-body ground-state solutions of the relativistic equations are obtained for various strengths of coupling, for both massive and massless mediating fields. The results show that the inclusion of virtual pairs has a large effect on the three-body binding energy at strong coupling.

Bergeron and Li have introduced a set of axioms which guarantee that the Grothendieck groups of a tower of algebras $\bigoplus_{n\ge0}A_n$ can be endowed with the structure of graded dual Hopf algebras. Hivert and Nzeutzhap, and independently Lam and Shimozono constructed dual graded graphs from primitive elements in Hopf algebras. In this paper we apply the composition of these constructions to towers of algebras. We show that if a tower $\bigoplus_{n\ge0}A_n$ gives rise to graded dual Hopf algebras then we must have $\dim(A_n)=r^nn!$ where $r = \dim(A_1)$. In the case $r=1$ we give a conjectural classification. We then investigate a quantum version of the main theorem. We conclude with some open problems and a categorification of the construction. This paper is a full version of the summary arXiv: 0710.3744., 18 pages. Minor changes, references updated

Given a list of $n$ cells $L=[(p_1,q_1),...,(p_n, q_n)]$ where $p_i, q_i\in \textbf{Z}_{\ge 0}$, we let $\Delta_L=\det |{(p_j!)^{-1}(q_j!)^{-1} x^{p_j}_iy^{q_j}_i} |$. The space of diagonally alternating polynomials is spanned by $\{\Delta_L\}$ where $L$ varies among all lists with $n$ cells. For $a>0$, the operators $E_a=\sum_{i=1}^{n} y_i\partial_{x_i}^a$ act on diagonally alternating polynomials and Haiman has shown that the space $A_n$ of diagonally alternating harmonic polynomials is spanned by $\{E_\lambda\Delta_n\}$. For $t=(t_m,...,t_1)\in \textbf{Z}_{> 0}^m$ with $t_m>...>t_1>0$, we consider here the operator $F_t=\det\big\|E_{t_{m-j+1}+(j-i)}\big\|$. Our first result is to show that $F_t\Delta_L$ is a linear combination of $\Delta_{L'}$ where $L'$ is obtained by {\sl moving} $\ell(t)=m$ distinct cells from $L$ in some determined fashion. This allows us to control the leading term of some elements of the form $F_{t_{(1)}}... F_{t_{(r)}}\Delta_n$. We use this to describe explicit bases of some of the bihomogeneous components of $A_n=\bigoplus A_n^{k,l}$ where $A_n^{k,l}=\hbox{Span}\{E_\lambda\Delta_n :\ell(\lambda)=l, |\lambda|=k\}$. More precisely we give an explicit basis of $A_n^{k,l}$ whenever $k, Comment: To appear in JCT-A; 21 pages, one PDF figure

Using the combinatorial techniques developed by Barcelo and Bergeron, we construct a Bn-module, called L (n), related to the Orlik-Solomon algebra of the hyperoctahedral hyperplane complements lattice (os(Bn)) [10, 13]. The Bn-modules L (n) and os(Bn) are analogous to the modules for the symmetric group which occur in the context of the free Lie algebra and the partition lattice. In particular we show that the module L (n) is the transpose of the module os(Bn) tensored by a sign representation. As a by-product we show that the action of Bn on a natural basis of L (n) is block triangular. The blocks are indexed by the conjugacy classes of Bn and have dimension equal to the number of elements in such a class. We also compute the characters of this action restricted to each block.

Bergeron and Li have introduced a set of axioms which guarantee that the Grothendieck groups of a tower of algebras $\bigoplus_{n \geq 0}A_n$ can be endowed with the structure of graded dual Hopf algebras. Hivert and Nzeutzhap, and independently Lam and Shimozono constructed dual graded graphs from primitive elements in Hopf algebras. In this paper we apply the composition of these constructions to towers of algebras. We show that if a tower $\bigoplus_{n \geq 0}A_n$ gives rise to graded dual Hopf algebras then we must have $\dim (A_n)=r^nn!$ where $r = \dim (A_1)$., Bergeron et Li ont donné un ensemble d'axiomes qui garanti que les groupes de Grothendieck d'une tour d'algèbres $\bigoplus_{n \geq 0}A_n$ peuvent être dotés d'une structure d'algèbres de Hopf graduées duales. Hivert et Nzeutzhap, et indépen\-damment Lam et Shimozono, ont construit des graphes gradués duals à partir d'éléments primitifs dans des algèbres de Hopf. Dans cet article, nous appliquons la composition de ces constructions aux tours des algèbres. Nous prouvons que si une tour $\bigoplus_{n \geq 0}A_n$ donne des algèbres de Hopf graduées duales, alors nous devons avoir $\dim (A_n)=r^nn!$ où $r = \dim (A_1)$.

We introduce non-commutative analogs of k-Schur functions of Lapointe–Lascoux and Morse. We give explicit formulas for the expansions of non-commutative functions with one and two parameters in terms of these new functions. These results are similar to the conjectures existing in the commutative case.

We prove that the pre-Lie operad is a free non-symmetric operad., Comment: 12 pages. Details on the trees used added in definition 1.2. The proof of corollary 3.6 has been changed