Sur une opérade ternaire liée 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 ) 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.