Sur une opérade ternaire liée aux treillis de Tamari

Frédéric Chapoton

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

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 20 (2011) no. 4, pp. 843-869.

Résumé
Abstract

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 ${(Y_{n})}_{n \geq 0}$ en construisant un isomorphisme entre $V (2 n + 1)$ et le groupe de Grothendieck de la catégorie $mod Y_{n}$ qui envoie la base de $V (2 n + 1)$ sur les classes des modules projectifs et qui transforme la structure anticyclique de $V$ en la transformation de Coxeter de la catégorie dérivée de $mod Y_{n}$ . La dualité de Koszul des opérades permet alors de calculer le polynôme caractéristique de cette transformation de Coxeter en utilisant une transformation de Legendre.

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 ${(Y_{n})}_{n \geq 0}$ by defining an isomorphism between $V (2 n + 1)$ and the Grothendieck group of the category $mod Y_{n}$ . This isomorphism maps the basis of $V (2 n + 1)$ to the classes of projective modules and sends the anticyclic map of the operad $V (2 n + 1)$ to the Coxeter transformation of the derived category of $mod Y_{n}$ . The Koszul duality theory for operads then allows us to compute the characteristic polynomial of the Coxeter transformation by a Legendre transform.

MR Zbl

DOI : 10.5802/afst.1326

Affiliations des auteurs :

Frédéric Chapoton ¹

¹ Institut Camille Jordan, Université Claude Bernard Lyon 1, Bâtiment Braconnier, 21 Avenue Claude Bernard, F-69622 Villeurbanne Cedex

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     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},
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     zbl = {1248.18009},
     language = {fr},
     url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1326/}
}

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Frédéric Chapoton. Sur une opérade ternaire liée aux treillis de Tamari. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 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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