Dual Creation Operators and a Dendriform
Algebra Structure on the Quasisymmetric
Functions

AbstractThe dual immaculate functions are a basis of the ring QSym of quasisymmetric functions and form one of the most natural analogues of the Schur functions. The dual immaculate function corresponding to a composition is a weighted generating function for immaculate tableaux in the same way as a Schur function is for semistandard Young tableaux; an immaculate tableau is defined similarly to a semistandard Young tableau, but the shape is a composition rather than a partition, and only the first column is required to strictly increase (whereas the other columns can be arbitrary, but each row has to weakly increase). Dual immaculate functions were introduced by Berg, Bergeron, Saliola, Serrano, and Zabrocki in arXiv:1208.5191, and have since been found to possess numerous nontrivial properties.In this note, we prove a conjecture of M. Zabrocki that provides an alternative construction for the dual immaculate functions in terms of certain “vertex operators”. The proof uses a dendriform structure on the ring QSym; we discuss the relation of this structure to known dendriformstructures on the combinatorial Hopf algebras FQSym andWQSym.

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Antipode Formulas for some Combinatorial Hopf Algebras

The Electronic Journal of Combinatorics ◽

10.37236/5949 ◽

2016 ◽

Vol 23 (4) ◽

Cited By ~ 2

Author(s):

Rebecca Patrias

Keyword(s):

Hopf Algebra ◽

Hopf Algebras ◽

Symmetric Functions ◽

Explicit Formulas ◽

Quasisymmetric Functions ◽

Noncommutative Symmetric Functions ◽

Combinatorial Hopf Algebras ◽

Combinatorial Formulas ◽

K Theory

Motivated by work of Buch on set-valued tableaux in relation to the K-theory of the Grassmannian, Lam and Pylyavskyy studied six combinatorial Hopf algebras that can be thought of as K-theoretic analogues of the Hopf algebras of symmetric functions, quasisymmetric functions, noncommutative symmetric functions, and of the Malvenuto-Reutenauer Hopf algebra of permutations. They described the bialgebra structure in all cases that were not yet known but left open the question of finding explicit formulas for the antipode maps. We give combinatorial formulas for the antipode map for the K-theoretic analogues of the symmetric functions, quasisymmetric functions, and noncommutative symmetric functions.

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A Uniform Generalization of Some Combinatorial Hopf Algebras

Algebras and Representation Theory ◽

10.1007/s10468-016-9648-x ◽

2016 ◽

Vol 20 (2) ◽

pp. 379-431 ◽

Cited By ~ 1

Author(s):

Jia Huang

Keyword(s):

Hopf Algebras ◽

Combinatorial Hopf Algebras

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Lie groups of controlled characters of combinatorial Hopf algebras

Annales de l’Institut Henri Poincaré D ◽

10.4171/aihpd/90 ◽

2020 ◽

Vol 7 (3) ◽

pp. 395-456 ◽

Cited By ~ 1

Author(s):

Rafael Dahmen ◽

Alexander Schmeding

Keyword(s):

Lie Groups ◽

Hopf Algebras ◽

Combinatorial Hopf Algebras

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Schubert Polynomials and $k$-Schur Functions

The Electronic Journal of Combinatorics ◽

10.37236/4139 ◽

2014 ◽

Vol 21 (4) ◽

Cited By ~ 1

Author(s):

Carolina Benedetti ◽

Nantel Bergeron

Keyword(s):

Finite Type ◽

Schur Functions ◽

Type A ◽

Bruhat Order ◽

Schur Function ◽

Quasisymmetric Functions ◽

Affine Grassmannian ◽

Schubert Polynomial ◽

Schubert Polynomials ◽

General Program

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 combinatorially 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. Then, 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. This is the first step of our more general program of showing combinatorially the positivity of the multiplication of a dual $k$-Schur function by a Schur function.

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Hopf Algebras from Branching Rules

Canadian Journal of Mathematics ◽

10.4153/cjm-1983-012-1 ◽

1983 ◽

Vol 35 (1) ◽

pp. 177-192 ◽

Cited By ~ 3

Author(s):

P. Hoffman

Keyword(s):

Abelian Group ◽

Symmetric Group ◽

Hopf Algebras ◽

Homogeneous Element ◽

Algebra Structure ◽

Graded Algebra ◽

Finitely Generated ◽

Dimension Zero ◽

Branching Rules

Below we work out the algebra structure of some Hopf algebras which arise concretely in restricting representations of the symmetric group to certain subgroups. The basic idea generalizes that used by Adams [1] for H*(BSU). The question arose in discussions with H. K. Farahat. I would like to thank him for his interest in the work and to acknowledge the usefulness of several stimulating conversations with him.1. Review and statement of results. A homogeneous element of a graded abelian group will have its gradation referred to as its dimension. In all such groups below there will be no non-zero elements with negative or odd dimension. A graded algebra (resp. coalgebra) will be associative (resp. coassociative), strictly commutative (resp. co-commutative) and in dimension zero will be isomorphic to the ground ring F, providing the unit (resp. counit). We shall deal amost entirely with F = Z or F = Z/p for a prime p; the cases F = 0 or a localization of Z will occur briefly. In every case, the component in each dimension will be a finitely generated free F-module, so dualization works simply.

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