Le module dendriforme sur le groupe cyclique

Chapoton, Frédéric

Annales de l'Institut Fourier, Tome 58 (2008) no. 7, pp. 2333-2350.

La structure d’opérade anticyclique de l’opérade dendriforme donne en particulier une matrice d’ordre $n$ agissant sur l’espace engendré par les arbres binaires plans à $n$ feuilles. On calcule le polynôme caractéristique de cette matrice. On propose aussi une conjecture compatible pour le polynôme caractéristique de la transformation de Coxeter du poset de Tamari, qui est essentiellement une racine carrée de cette matrice.

It is known that the Dendriform operad is in fact an anticyclic operad. This refined structure defines in particular a matrix of finite order acting on the vector space spanned by planar binary trees. We compute here its characteristic polynomial and propose a compatible conjecture for the characteristic polynomial of the Coxeter transformation for the Tamari lattice, which is essentially a square root of this matrix.

[1] Aguiar, M.; Sottile, F. Structure of the Loday-Ronco Hopf algebra of trees, J. Algebra, Volume 295 (2006) no. 2, pp. 473-511 | MR | Zbl

[2] Chapoton, F. On some anticyclic operads, Algebr. Geom. Topol., Volume 5 (2005), p. 53-69 (electronic) | DOI | MR | Zbl

[3] Chapoton, F. The anticyclic operad of moulds, Int. Math. Res. Not. IMRN (2007) no. 20, pp. Art. ID rnm078, 36 | MR

[4] Chapoton, Frédéric On the Coxeter transformations for Tamari posets, Canad. Math. Bull., Volume 50 (2007) no. 2, pp. 182-190 | DOI | MR

[5] Curtis, C. W.; Reiner, I. Representation theory of finite groups and associative algebras, Pure and Applied Mathematics, Vol. XI, Interscience Publishers, a division of John Wiley & Sons, New York-London, 1962 | MR | Zbl

[6] Hivert, F.; Novelli, J.-C.; Thibon, J.-Y. Un analogue du monoïde plaxique pour les arbres binaires de recherche, C. R. Math. Acad. Sci. Paris, Volume 335 (2002) no. 7, pp. 577-580 | MR | Zbl

[7] Hivert, F.; Novelli, J.-C.; Thibon, J.-Y. Sur quelques propriétés de l’algèbre des arbres binaires, C. R. Math. Acad. Sci. Paris, Volume 337 (2003) no. 9, pp. 565-568 | Zbl

[8] Hivert, F.; Novelli, J.-C.; Thibon, J.-Y. The algebra of binary search trees, Theoret. Comput. Sci., Volume 339 (2005) no. 1, pp. 129-165 | DOI | MR | Zbl

[9] Huang, S.; Tamari, D. Problems of associativity : A simple proof for the lattice property of systems ordered by a semi-associative law, J. Combinatorial Theory Ser. A, Volume 13 (1972), pp. 7-13 | DOI | MR | Zbl

[10] Loday, J.-L. Dialgebras, Dialgebras and related operads (Lecture Notes in Math.), Volume 1763, Springer, Berlin, 2001, pp. 7-66 | MR | Zbl

[11] Loday, J.-L. Arithmetree, J. Algebra, Volume 258 (2002) no. 1, pp. 275-309 (Special issue in celebration of Claudio Procesi’s 60th birthday) | DOI | MR | Zbl

[12] Loday, J.-L.; Ronco, M. O. Hopf algebra of the planar binary trees, Adv. Math., Volume 139 (1998) no. 2, pp. 293-309 | DOI | MR | Zbl

[13] Loday, J.-L.; Ronco, M. O. Order structure on the algebra of permutations and of planar binary trees, J. Algebraic Combin., Volume 15 (2002) no. 3, pp. 253-270 | DOI | MR | Zbl

[14] Macdonald, I. G. Symmetric functions and Hall polynomials, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 1995 (With contributions by A. Zelevinsky, Oxford Science Publications) | MR | Zbl