We introduce an anticyclic operad 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 by defining an isomorphism between and the Grothendieck group of the category . This isomorphism maps the basis of to the classes of projective modules and sends the anticyclic map of the operad to the Coxeter transformation of the derived category of . 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 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 en construisant un isomorphisme entre et le groupe de Grothendieck de la catégorie qui envoie la base de sur les classes des modules projectifs et qui transforme la structure anticyclique de en la transformation de Coxeter de la catégorie dérivée de . 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 = {Fr\'ed\'eric Chapoton},
title = {Sur une op\'erade ternaire li\'ee aux treillis de {Tamari}},
journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
pages = {843--869},
publisher = {Universit\'e Paul Sabatier, Institut de Math\'ematiques},
address = {Toulouse},
volume = {6e s{\'e}rie, 20},
number = {4},
year = {2011},
doi = {10.5802/afst.1326},
mrnumber = {2918216},
zbl = {1248.18009},
language = {fr},
url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1326/}
}
TY - JOUR AU - Frédéric Chapoton TI - Sur une opérade ternaire liée aux treillis de Tamari JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2011 SP - 843 EP - 869 VL - 20 IS - 4 PB - Université Paul Sabatier, Institut de Mathématiques PP - Toulouse UR - https://afst.centre-mersenne.org/articles/10.5802/afst.1326/ DO - 10.5802/afst.1326 LA - fr ID - AFST_2011_6_20_4_843_0 ER -
%0 Journal Article %A Frédéric Chapoton %T Sur une opérade ternaire liée aux treillis de Tamari %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2011 %P 843-869 %V 20 %N 4 %I Université Paul Sabatier, Institut de Mathématiques %C Toulouse %U https://afst.centre-mersenne.org/articles/10.5802/afst.1326/ %R 10.5802/afst.1326 %G fr %F AFST_2011_6_20_4_843_0
Frédéric Chapoton. 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. https://afst.centre-mersenne.org/articles/10.5802/afst.1326/
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