Hopf algebras and dendriform structures arising from parking functions

Jean-Christophe Novelli ¹   ; Jean-Yves Thibon ¹

¹ Institut Gaspard Monge Université de Marne-la-Vallée 5 Boulevard Descartes Champs-sur-Marne 77454 Marne-la-Vallée Cedex 2, France

Fundamenta Mathematicae, Tome 193 (2007) no. 3, pp. 189-241

Résumé

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 occurring 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 a left ideal.

Détail

DOI : 10.4064/fm193-3-1

Keywords: introduce graded hopf algebra based set parking functions hence dimension degree algebra embedded noncommutative polynomial algebra infinitely many variables determine its structure admits natural quotients subalgebras whose graded components have dimensions respectively given schr der numbers plane trees catalan numbers powers these smaller algebras always bialgebras belong family di trialgebras occurring works loday ronco moreover fundamental notion parkization allows endow set parking functions fixed length associative multiplication different coming shi arrangement leading generalization internal product symmetric functions several intermediate algebras stable under operation among finds solomon descent algebra algebra based catalan set admitting solomon algebra ideal