Hopf algebras of symmetric functions and class functions

1977 ◽  
pp. 168-181 ◽  
Author(s):  
Ladnor Geissinger
Keyword(s):  
Hopf Algebras ◽  
Symmetric Functions

scholarly journals Antipode Formulas for some Combinatorial Hopf Algebras

10.37236/5949 ◽  
2016 ◽  
Vol 23 (4) ◽  
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.

scholarly journals Combinatorial Hopf Algebras of Simplicial Complexes

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Carolina Benedetti ◽  
Joshua Hallam ◽  
John Machacek
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Symmetric Functions ◽  
Simplicial Complexes ◽  
Combinatorial Hopf Algebras ◽  
International Audience ◽  
Donnent Lieu

International audience We consider a Hopf algebra of simplicial complexes and provide a cancellation-free formula for its antipode. We then obtain a family of combinatorial Hopf algebras by defining a family of characters on this Hopf algebra. The characters of these Hopf algebras give rise to symmetric functions that encode information about colorings of simplicial complexes and their $f$-vectors. We also use characters to give a generalization of Stanley’s $(-1)$-color theorem. Nous considérons une algèbre de Hopf de complexes simpliciaux et fournissons une formule sans multiplicité pour son antipode. On obtient ensuite une famille d'algèbres de Hopf combinatoires en définissant une famille de caractères sur cette algèbre de Hopf. Les caractères de ces algèbres de Hopf donnent lieu à des fonctions symétriques qui encode de l’information sur les coloriages du complexe simplicial ainsi que son vecteur-$f$. Nousallons également utiliser des caractères pour donner une généralisation du théorème $(-1)$ de Stanley.

scholarly journals Supercharacters, symmetric functions in noncommuting variables (extended abstract)

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Marcelo Aguiar ◽  
Carlos André ◽  
Carolina Benedetti ◽  
Nantel Bergeron ◽  
Zhi Chen ◽  
...  
Keyword(s):  
Hopf Algebra ◽  
Fourier Analysis ◽  
Finite Field ◽  
Hopf Algebras ◽  
Symmetric Functions ◽  
Combinatorial Hopf Algebras ◽  
Nous Montrons ◽  
International Audience

International audience 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.

scholarly journals Weak quasi-symmetric functions, Rota–Baxter algebras and Hopf algebras

2019 ◽  
Vol 344 ◽  
pp. 1-34 ◽  
Author(s):  
Houyi Yu ◽  
Li Guo ◽  
Jean-Yves Thibon
Keyword(s):  
Hopf Algebras ◽  
Symmetric Functions

scholarly journals The Heisenberg product: from Hopf algebras and species to symmetric functions

2017 ◽  
Vol 11 (2) ◽  
pp. 261-311 ◽  
Author(s):  
Marcelo Aguiar ◽  
Walter Ferrer Santos ◽  
Walter Moreira
Keyword(s):  
Hopf Algebras ◽  
Symmetric Functions

scholarly journals Polytopes, Fibonacci numbers, Hopf algebras, and quasi-symmetric functions

2011 ◽  
Vol 66 (2) ◽  
pp. 271-367 ◽  
Author(s):  
Viktor M Buchstaber ◽  
Nikolai Yu Erokhovets
Keyword(s):  
Hopf Algebras ◽  
Symmetric Functions ◽  
Fibonacci Numbers

scholarly journals Hopf Algebra Structure of Generalized Quasi-Symmetric Functions in Partially Commutative Variables

2021 ◽  
Vol 28 (2) ◽  
Author(s):  
Adam Doliwa
Keyword(s):  
Power Series ◽  
Hopf Algebra ◽  
Hopf Algebras ◽  
Symmetric Functions ◽  
Algebra Structure ◽  
Natural Basis ◽  
Dual Algebra ◽  
Power Sum ◽  
Hopf Algebra Structure

We introduce a coloured generalization  $\mathrm{NSym}_A$ of the Hopf algebra of non-commutative symmetric functions  described as a subalgebra of the of rooted ordered coloured trees Hopf algebra. Its natural basis can be identified with the set of sentences over alphabet $A$ (the set of colours). We present also its graded dual algebra $\mathrm{QSym}_A$ of coloured quasi-symmetric functions together with its realization in terms of power series in partially commutative variables.  We provide formulas expressing multiplication, comultiplication and the antipode for these Hopf algebras in various bases — the corresponding generalizations of the complete homogeneous, elementary, ribbon Schur and power sum bases of $\mathrm{NSym}$, and the monomial and fundamental bases of $\mathrm{QSym}$. We study also certain distinguished series of trees in the setting of restricted duals to Hopf algebras.

scholarly journals Supercharacters, symmetric functions in noncommuting variables, and related Hopf algebras

2012 ◽  
Vol 229 (4) ◽  
pp. 2310-2337 ◽  
Author(s):  
Marcelo Aguiar ◽  
Carlos André ◽  
Carolina Benedetti ◽  
Nantel Bergeron ◽  
Zhi Chen ◽  
...  
Keyword(s):  
Hopf Algebras ◽  
Symmetric Functions
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