scholarly journals Hopf algebras of planar binary trees: an operated algebra approach

2019 ◽  
Vol 51 (4) ◽  
pp. 567-588 ◽  
Author(s):  
Yi Zhang ◽  
Xing Gao
Keyword(s):  
Hopf Algebras ◽  
Binary Trees ◽  
Algebra Approach ◽  
Planar Binary Trees

scholarly journals QED Hopf algebras on planar binary trees

2003 ◽  
Vol 267 (1) ◽  
pp. 298-322 ◽  
Author(s):  
Christian Brouder ◽  
Alessandra Frabetti
Keyword(s):  
Hopf Algebras ◽  
Binary Trees ◽  
Planar Binary Trees

scholarly journals Brick polytopes, lattices and Hopf algebras

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Vincent Pilaud
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Transitive Closure ◽  
Weak Order ◽  
Binary Trees ◽  
International Audience ◽  
Congruence Classes

International audience Generalizing the connection between the classes of the sylvester congruence and the binary trees, we show that the classes of the congruence of the weak order on Sn defined as the transitive closure of the rewriting rule UacV1b1 ···VkbkW ≡k UcaV1b1 ···VkbkW, for letters a < b1,...,bk < c and words U,V1,...,Vk,W on [n], are in bijection with acyclic k-triangulations of the (n + 2k)-gon, or equivalently with acyclic pipe dreams for the permutation (1,...,k,n + k,...,k + 1,n + k + 1,...,n + 2k). It enables us to transport the known lattice and Hopf algebra structures from the congruence classes of ≡k to these acyclic pipe dreams, and to describe the product and coproduct of this algebra in terms of pipe dreams. Moreover, it shows that the fan obtained by coarsening the Coxeter fan according to the classes of ≡k is the normal fan of the corresponding brick polytope

scholarly journals Hopf Algebra of the Planar Binary Trees

1998 ◽  
Vol 139 (2) ◽  
pp. 293-309 ◽  
Author(s):  
Jean-Louis Loday ◽  
María O. Ronco
Keyword(s):  
Hopf Algebra ◽  
Binary Trees ◽  
Planar Binary Trees

Topologies of locally minimal planar binary trees

1994 ◽  
Vol 49 (6) ◽  
pp. 201-202
Author(s):  
A O Ivanov ◽  
A A Tuzhilin
Keyword(s):  
Binary Trees ◽  
Planar Binary Trees

scholarly journals Lattice of combinatorial Hopf algebras: binary trees with multiplicities

2013 ◽  
Vol DMTCS Proceedings vol. AS,... (Proceedings) ◽  
Author(s):  
Jean-Baptiste Priez
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Lattice Structure ◽  
Binary Trees ◽  
Hook Length ◽  
Combinatorial Hopf Algebras ◽  
Combinatorial Hopf Algebra ◽  
International Audience ◽  
Comme Application ◽  
Premiere Partie

International audience In a first part, we formalize the construction of combinatorial Hopf algebras from plactic-like monoids using polynomial realizations. Thank to this construction we reveal a lattice structure on those combinatorial Hopf algebras. As an application, we construct a new combinatorial Hopf algebra on binary trees with multiplicities and use it to prove a hook length formula for those trees. Dans une première partie, nous formalisons la construction d’algèbres de Hopf combinatoires à partir d’une réalisation polynomiale et de monoïdes de type monoïde plaxique. Grâce à cette construction, nous mettons à jour une structure de treillis sur ces algèbres de Hopf combinatoires. Comme application, nous construisons une nouvelle algèbre de Hopf sur des arbres binaires à multiplicités et on l’utilise pour démontrer une formule des équerressur ces arbres.

scholarly journals New Hopf Structures on Binary Trees

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Stefan Forcey ◽  
Aaron Lauve ◽  
Frank Sottile
Keyword(s):  
Hopf Algebras ◽  
Homotopy Theory ◽  
Algebraic Combinatorics ◽  
Binary Trees ◽  
Independent Interest ◽  
Hopf Module ◽  
Möbius Inversion ◽  
International Audience ◽  
Higher Categories ◽  
Donne Lieu

International audience The multiplihedra $\mathcal{M}_{\bullet} = (\mathcal{M}_n)_{n \geq 1}$ form a family of polytopes originating in the study of higher categories and homotopy theory. While the multiplihedra may be unfamiliar to the algebraic combinatorics community, it is nestled between two families of polytopes that certainly are not: the permutahedra $\mathfrak{S}_{\bullet}$ and associahedra $\mathcal{Y}_{\bullet}$. The maps $\mathfrak{S}_{\bullet} \twoheadrightarrow \mathcal{M}_{\bullet} \twoheadrightarrow \mathcal{Y}_{\bullet}$ reveal several new Hopf structures on tree-like objects nestled between the Hopf algebras $\mathfrak{S}Sym$ and $\mathcal{Y}Sym$. We begin their study here, showing that $\mathcal{M}Sym$ is a module over $\mathfrak{S}Sym$ and a Hopf module over $\mathcal{Y}Sym$. An elegant description of the coinvariants for $\mathcal{M}Sym$ over $\mathcal{Y}Sym$ is uncovered via a change of basis-using Möbius inversion in posets built on the $1$-skeleta of $\mathcal{M}_{\bullet}$. Our analysis uses the notion of an $\textit{interval retract}$ that should be of independent interest in poset combinatorics. It also reveals new families of polytopes, and even a new factorization of a known projection from the associahedra to hypercubes. Les multiplièdres $\mathcal{M}_{\bullet} = (\mathcal{M}_n)_{n \geq 1}$ forment une famille de polytopes en provenant de l'étude des catégories supérieures et de la théorie de l'homotopie. Tandis que les multiplihèdres sont peu connus dans la communauté de la combinatoire algébrique, ils sont nichés entre deux familles des polytopes qui sont bien connus: les permutahèdres $\mathfrak{S}_{\bullet}$ et les associahèdres $\mathcal{Y}_{\bullet}$. Les morphismes $\mathfrak{S}_{\bullet} \twoheadrightarrow \mathcal{M}_{\bullet} \twoheadrightarrow \mathcal{Y}_{\bullet}$ dévoilent plusieurs nouvelles structures de Hopf sur les arbres binaires entre les algèbres de Hopf $\mathfrak{S}Sym$ et $\mathcal{Y}Sym$. Nous commençons son étude ici, en démontrant que $\mathcal{M}Sym$ est un module sur $\mathfrak{S}Sym$ et un module de Hopf sur $\mathcal{Y}Sym$. Une description élégante des coinvariants de $\mathcal{M}Sym$ sur $\mathcal{Y}Sym$ est trouvée par moyen d'une change de base―en utilisant une inversion de Möbius dans certains posets construits sur le $1$-squelette de $\mathcal{M}_{\bullet}$. Notre analyse utilise la notion d'$\textit{interval retract}$, qui devrait être intéressante par soi-même dans la théorie des ensembles partiellement ordonnés. Notre analyse donne lieu également à des nouvelles familles des polytopes, et même une nouvelle factorisation d'une projection connue des associahèdres aux hypercubes.

Hopf Algebras and Projective Representations of G ≀ Sn AND G ≀ An

1986 ◽  
Vol 38 (6) ◽  
pp. 1380-1458 ◽  
Author(s):  
Peter N. Hoffman ◽  
John F. Humphreys
Keyword(s):  
Hopf Algebras ◽  
Unsolved Problem ◽  
Alternating Group ◽  
Induced Representations ◽  
Linear Representations ◽  
Algebra Approach ◽  
Projective Representations ◽  
New Formulation ◽  
Complex Projective ◽  
Extra Structure

In 1911, Schur published a rather formidable paper [9] in which he determined all the complex projective characters for the symmetric group (denoted Σn here, despite the title), and for the alternating group An (A pronounced “alpha”). As far as we know, the construction of the modules involved is still an unsolved problem. The results of Schur can be expressed in terms of certain induced representations whose characters form a basis for the group of virtual characters, plus formulae expressing the irreducible characters in terms of these induced characters. Here we give a new formulation of the above induced characters in the spirit of the well known “induction algebra” approach to the linear representations of Σn. We use some Hopf algebra techniques inspired by [5] to give new proofs of Schur's results, and to determine the extra structure which we define.

scholarly journals Renormalization of QED with planar binary trees

2001 ◽  
Vol 19 (4) ◽  
pp. 715-741 ◽  
Author(s):  
Ch. Brouder ◽  
A. Frabetti
Keyword(s):  
Binary Trees ◽  
Planar Binary Trees

scholarly journals Superization and $(q,t)$-Specialization in Combinatorial Hopf Algebras

10.37236/87 ◽  
2009 ◽  
Vol 16 (2) ◽  
Author(s):  
Jean-Christophe Novelli ◽  
Jean-Yves Thibon
Keyword(s):  
Hopf Algebras ◽  
Symmetric Functions ◽  
Binary Trees ◽  
Hook Length ◽  
Combinatorial Hopf Algebras ◽  
Plane Trees ◽  
Classical Construction

We extend a classical construction on symmetric functions, the superization process, to several combinatorial Hopf algebras, and obtain analogs of the hook-content formula for the $(q,t)$-specializations of various bases. Exploiting the dendriform structures yields in particular $(q,t)$-analogs of the Björner-Wachs $q$-hook-length formulas for binary trees, and similar formulas for plane trees.

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