Le module dendriforme sur le groupe cyclique

TL;DR

This paper investigates the algebraic structure of the dendriform operad related to cyclic groups, computes its characteristic polynomial, and conjectures a related polynomial for the Tamari lattice's Coxeter transformation.

Contribution

It provides the first explicit computation of the characteristic polynomial of the matrix associated with the dendriform operad and proposes a conjecture linking it to the Tamari lattice.

Findings

Computed the characteristic polynomial of the matrix on the dendriform operad.

Proposed a conjecture for the characteristic polynomial of the Coxeter transformation.

Linked the matrix's polynomial to the Tamari lattice's structure.

Abstract

The structure of anticyclic operad on the Dendriform operad defines in particular a matrix of finite order acting on the vector space spanned by planar binary trees. We compute its characteristic polynomial and propose a (compatible) conjecture for the characteristic polynomial of the Coxeter transformation for the Tamari lattice, which is mostly a square root of this matrix.