Multiple Lie Derivatives and Forests

Florent Hivert (1) , Nefton Pali (2)

1 GALaC - Graphes, Algorithmes et Combinatoire - LISN
2 LM-Orsay - Laboratoire de Mathématiques d'Orsay

We obtain a complete time expansion of the pull-back operator generated by a real analytic flow of real analytic automorphisms acting on analytic tensor sections of a manifold. Our expansion is given in terms of multiple Lie derivatives. Motivated by this expansion, we provide a rather simple and explicit estimate for higher order covariant derivatives of multiple Lie derivatives acting on smooth endomorphism sections of the tangent bundle of a manifold. We assume the covariant derivative to be torsion free. The estimate is given in terms of Dyck polynomials. The proof uses a new result on the combinatorics of rooted labeled ordered forests and Dyck polynomials.

Domaines: Combinatoire [math.CO]; Géométrie différentielle [math.DG]

Florent Hivert, Nefton Pali. Multiple Lie Derivatives and Forests. Advances in Mathematics, 2019, 354, ⟨10.1016/j.aim.2019.106732⟩. ⟨hal-02349044⟩