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arXiv · 1303.5538

A lattice of combinatorial Hopf algebras, Application to binary trees with multiplicities

Jean-Baptiste Priez
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Abstract

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 premi\`ere partie, nous formalisons la construction d'alg\`ebres de Hopf combinatoires \`a partir d'une r\'ealisation polynomiale et de mono\"ides de type mono\"ide plaxique. Gr\^ace \`a cette construction, nous mettons \`a jour une structure de treillis sur ces alg\`ebres de Hopf combinatoires. Comme application, nous construisons une nouvelle alg\`ebre de Hopf sur des arbres binaires \`a multiplicit\'es et on l'utilise pour d\'emontrer une formule des \'equerres sur ces arbres.

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Jean-Baptiste Priez. 2013-03-22. A lattice of combinatorial Hopf algebras, Application to binary trees with multiplicities. https://arxiv.org/abs/1303.5538

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