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

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.

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.

@article{AFST_2011_6_20_4_843_0,
     author = {Chapoton, Fr\'ed\'eric},
     title = {Sur une op\'erade ternaire li\'ee aux treillis de Tamari},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {6e s{\'e}rie, 20},
     number = {4},
     year = {2011},
     pages = {843-869},
     doi = {10.5802/afst.1326},
     mrnumber = {2918216},
     zbl = {1248.18009},
     language = {fr},
     url = {http://www.numdam.org/item/AFST_2011_6_20_4_843_0}
}
Chapoton, Frédéric. Sur une opérade ternaire liée aux treillis de Tamari. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 20 (2011) no. 4, pp. 843-869. doi : 10.5802/afst.1326. http://www.numdam.org/item/AFST_2011_6_20_4_843_0/

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