Sur une opérade ternaire liée aux treillis de Tamari
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 20 (2011) no. 4, pp. 843-869.

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 ) n0 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 ) n0 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.

DOI : 10.5802/afst.1326

Frédéric Chapoton 1

1 Institut Camille Jordan, Université Claude Bernard Lyon 1, Bâtiment Braconnier, 21 Avenue Claude Bernard, F-69622 Villeurbanne Cedex
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Frédéric Chapoton. Sur une opérade ternaire liée aux treillis de Tamari. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 20 (2011) no. 4, pp. 843-869. doi : 10.5802/afst.1326. https://afst.centre-mersenne.org/articles/10.5802/afst.1326/

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