Le module dendriforme sur le groupe cyclique

Chapoton, Frédéric

Annales de l'Institut Fourier, Tome 58 (2008) no. 7, pp. 2333-2350.

La structure d’opérade anticyclique de l’opérade dendriforme donne en particulier une matrice d’ordre $n$ agissant sur l’espace engendré par les arbres binaires plans à $n$ feuilles. On calcule le polynôme caractéristique de cette matrice. On propose aussi une conjecture compatible pour le polynôme caractéristique de la transformation de Coxeter du poset de Tamari, qui est essentiellement une racine carrée de cette matrice.

It is known that the Dendriform operad is in fact an anticyclic operad. This refined structure defines in particular a matrix of finite order acting on the vector space spanned by planar binary trees. We compute here its characteristic polynomial and propose a compatible conjecture for the characteristic polynomial of the Coxeter transformation for the Tamari lattice, which is essentially a square root of this matrix.

Classification : 18D50, 05E05, 06A07
Mot clés : opérade anticyclique, opérade dendriforme, treillis de Tamari, transformation de Coxeter
Keywords: Dendriform operad, anticyclic operad, Tamari lattice, Coxeter transformation

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