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- Série 6 : Tome 20 (2011)

- no. 4

- p. 843-869

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On introduit une opérade anticyclique V définie par une présentation ternaire quadratique. On montre qu’elle admet une base indexée par les arbres binaires planaires. On relie cette construction à la famille des treillis de Tamari (Y n ) n≥0 en construisant un isomorphisme entre V(2n+1) et le groupe de Grothendieck de la catégorie modY n qui envoie la base de V(2n+1) sur les classes des modules projectifs et qui transforme la structure anticyclique de V en la transformation de Coxeter de la catégorie dérivée de modY n . La dualité de Koszul des opérades permet alors de calculer le polynôme caractéristique de cette transformation de Coxeter en utilisant une transformation de Legendre.

We introduce an anticyclic operad V given by a ternary generator and a quadratic relation. We show that it admits a natural basis indexed by planar binary trees. We then relate this construction to the familly of Tamari lattices (Y n ) n≥0 by defining an isomorphism between V(2n+1) and the Grothendieck group of the category modY n . This isomorphism maps the basis of V(2n+1) to the classes of projective modules and sends the anticyclic map of the operad V(2n+1) to the Coxeter transformation of the derived category of modY n . The Koszul duality theory for operads then allows us to compute the characteristic polynomial of the Coxeter transformation by a Legendre transform.

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Frédéric Chapoton 1

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