Sur une op\'erade ternaire li\'ee aux treillis de Tamari

Abstract

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) for n>=0 by defining an isomorphism between V(2n+1) and the Grothendieck group of the category mod Y_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 to the Coxeter transformation of the derived category of mod Y_n. One can then use Koszul duality and a Legendre transform to compute the characteristic polynomial of these Coxeter transformations.