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Quasishuffle products
 J. Algebraic Combin
"... Abstract. Given a locally finite graded set A and a commutative, associative operation on A that adds degrees, we construct a commutative multiplication ∗ on the set of noncommutative polynomials in A which we call a quasishuffle product; it can be viewed as a generalization of the shuffle product ..."
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Abstract. Given a locally finite graded set A and a commutative, associative operation on A that adds degrees, we construct a commutative multiplication ∗ on the set of noncommutative polynomials in A which we call a quasishuffle product; it can be viewed as a generalization of the shuffle product x. We extend this commutative algebra structure to a Hopf algebra (A, ∗, ∆); in the case where A is the set of positive integers and the operation on A is addition, this gives the Hopf algebra of quasisymmetric functions. If rational coefficients are allowed, the quasishuffle product is in fact no more general than the shuffle product; we give an isomorphism exp of the shuffle Hopf algebra (A,x,∆) onto (A, ∗, ∆). Both the set L of Lyndon words on A and their images {exp(w)  w ∈ L} freely generate the algebra (A, ∗). We also consider the graded dual of (A, ∗,∆). We define a deformation ∗q of ∗ that coincides with ∗ when q = 1 and is isomorphic to the concatenation product when q is not a root of unity. Finally, we discuss various examples, particularly the algebra of quasisymmetric functions (dual to the noncommutative symmetric functions) and the algebra of Euler sums.
Combinatorial Hopf algebras and generalized DehnSommerville relations
, 2003
"... A combinatorial Hopf algebra is a graded connected Hopf algebra over a field k equipped with a character (multiplicative linear functional) ζ: H → k. We show that the terminal object in the category of combinatorial Hopf algebras is the algebra QSym of quasisymmetric functions; this explains the u ..."
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Cited by 95 (20 self)
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A combinatorial Hopf algebra is a graded connected Hopf algebra over a field k equipped with a character (multiplicative linear functional) ζ: H → k. We show that the terminal object in the category of combinatorial Hopf algebras is the algebra QSym of quasisymmetric functions; this explains the ubiquity of quasisymmetric 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, ζ) possesses two canonical Hopf subalgebras on which the character ζ is even (respectively, odd). The odd subalgebra is defined by certain canonical relations which we call the generalized DehnSommerville relations. We show that, for H = QSym, the generalized DehnSommerville relations are the BayerBillera 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 MalvenutoReutenauer Hopf algebra of permutations, the
Order structure on the algebra of permutations and of planar binary trees
 J. Alg. Combin
"... Abstract. Let Xn be either the symmetric group on n letters, the set of planar binary ntrees or the set of vertices of the (n − 1)dimensional cube. In each case there exists a graded associative product on ⊕ n≥0 K [Xn]. We prove that it can be described explicitly by using the weak Bruhat order on ..."
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Cited by 35 (5 self)
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Abstract. Let Xn be either the symmetric group on n letters, the set of planar binary ntrees or the set of vertices of the (n − 1)dimensional cube. In each case there exists a graded associative product on ⊕ n≥0 K [Xn]. We prove that it can be described explicitly by using the weak Bruhat order on Sn, the lefttoright order on planar trees, the lexicographic order in the cube case. Keywords: planar binary tree, order structure, weak Bruhat order, algebra of permutations, dendriform algebra
Lattice congruences, fans and Hopf algebras
 J. Combin. Theory Ser. A
"... Abstract. We give a unified explanation of the geometric and algebraic properties of two wellknown maps, one from permutations to triangulations, and another from permutations to subsets. Furthermore we give a broad generalization of the maps. Specifically, for any lattice congruence of the weak or ..."
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Cited by 32 (12 self)
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Abstract. We give a unified explanation of the geometric and algebraic properties of two wellknown maps, one from permutations to triangulations, and another from permutations to subsets. Furthermore we give a broad generalization of the maps. Specifically, for any lattice congruence of the weak order on a Coxeter group we construct a complete fan of convex cones with strong properties relative to the corresponding lattice quotient of the weak order. We show that if a family of lattice congruences on the symmetric groups satisfies certain compatibility conditions then the family defines a sub Hopf algebra of the MalvenutoReutenauer Hopf algebra of permutations. Such a sub Hopf algebra has a basis which is described by a type of patternavoidance. Applying these results, we build the MalvenutoReutenauer algebra as the limit of an infinite sequence of smaller algebras, where the second algebra in the sequence is the Hopf algebra of noncommutative symmetric functions. We also associate both a fan and a Hopf algebra to a set of permutations which appears to be equinumerous with the Baxter permutations. 1.
A primer of Hopf algebras
 Insitut des Hautes Études Scientifiques, IHES/M/06/04
, 2006
"... Summary. In this paper, we review a number of basic results about socalled Hopf algebras. We begin by giving a historical account of the results obtained in the 1930’s and 1940’s about the topology of Lie groups and compact symmetric spaces. The climax is provided by the structure theorems due to H ..."
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Summary. In this paper, we review a number of basic results about socalled Hopf algebras. We begin by giving a historical account of the results obtained in the 1930’s and 1940’s about the topology of Lie groups and compact symmetric spaces. The climax is provided by the structure theorems due to Hopf, Samelson, Leray and Borel. The main part of this paper is a thorough analysis of the relations between Hopf algebras and Lie groups (or algebraic groups). We emphasize especially the category of unipotent (and prounipotent) algebraic groups, in connection with MilnorMoore’s theorem. These methods are a powerful tool to show that some algebras are free polynomial rings. The last part is an introduction to the combinatorial aspects of polylogarithm functions and the corresponding multiple zeta values. 1 Introduction.............................................
Noncommutative Pieri Operators On Posets
 J. Combin. Th. Ser. A
, 2000
"... We consider graded representations of the algebra NC of noncommutative symmetric functions on the Zlinear span of a graded poset P. The matrix coefficients of such a representation give a Hopf morphism from a Hopf algebra generated by the intervals of P to the Hopf algebra of quasisymmetric fun ..."
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Cited by 29 (5 self)
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We consider graded representations of the algebra NC of noncommutative symmetric functions on the Zlinear span of a graded poset P. The matrix coefficients of such a representation give a Hopf morphism from a Hopf algebra generated by the intervals of P to the Hopf algebra of quasisymmetric functions.
HOPF ALGEBRAS AND DENDRIFORM STRUCTURES ARISING FROM PARKING FUNCTIONS
, 2005
"... We introduce a graded Hopf algebra based on the set of parking functions (hence of dimension (n + 1) n−1 in degree n). This algebra can be embedded into a noncommutative polynomial algebra in infinitely many variables. We determine its structure, and show that it admits natural quotients and subalge ..."
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Cited by 27 (12 self)
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We introduce a graded Hopf algebra based on the set of parking functions (hence of dimension (n + 1) n−1 in degree n). This algebra can be embedded into a noncommutative polynomial algebra in infinitely many variables. We determine its structure, and show that it admits natural quotients and subalgebras whose graded components have dimensions respectively given by the Schröder numbers (plane trees), the Catalan numbers, and powers of 3. These smaller algebras are always bialgebras and belong to some family of di or trialgebras occuring in the works of Loday and Ronco. Moreover, the fundamental notion of parkization allows one to endow the set of parking functions of fixed length with an associative multiplication (different from the one coming from the Shi arrangement), leading to a generalization of the internal product of symmetric functions. Several of the intermediate algebras are stable under this operation. Among them, one finds the Solomon descent algebra but also a new algebra based on a Catalan set, admitting the Solomon algebra as
Generalized Riffle Shuffles and Quasisymmetric Functions
, 2001
"... Let x i be a probability distribution on a totally ordered set I, i.e., the probability of i 2 I is x i . (Hence x i 0 and x i = 1.) Fix n 2 P = f1; 2; : : :g, and define a random permutation w 2 S n as follows. For each 1 j n, choose independently an integer i j (from the distribution x i ). ..."
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Cited by 27 (0 self)
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Let x i be a probability distribution on a totally ordered set I, i.e., the probability of i 2 I is x i . (Hence x i 0 and x i = 1.) Fix n 2 P = f1; 2; : : :g, and define a random permutation w 2 S n as follows. For each 1 j n, choose independently an integer i j (from the distribution x i ). Then standardize the sequence i = i 1 \Delta \Delta \Delta i n in the sense of [34, p. 322], i.e., let ff 1 ! \Delta \Delta \Delta ! ff k be the elements of I actually appearing in i, and let a i be the number of ff i 's in i. Replace the ff 1 's in i by 1; 2; : : : ; a 1 from lefttoright, then the ff 2 's in i by a 1 + 1; a 1 + 2; : : : ; a 1 + a 2 from lefttoright, etc. For instance, if I = P and i = 311431, then w = 412653. This defines a probability distribution on the symmetric group S n , which we call the QSdistribution (because of the close connection with quasisymmetric functions explained below). If we need to be explicit about the parameters x = (x i ) i2I , t