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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.

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.

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
     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/}
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
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%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
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%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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