Rooted tree maps and the Kawashima relations for multiple zeta values

TL;DR

This paper demonstrates that the linear part of Kawashima relations for multiple zeta values can be derived from rooted tree maps, connecting combinatorial structures with algebraic relations.

Contribution

It proves that rooted tree map relations imply the linear Kawashima relations, establishing a reverse implication and deepening the understanding of multiple zeta value relations.

Findings

Rooted tree maps generate relations among multiple zeta values.

The linear Kawashima relations are implied by rooted tree map relations.

Establishes a connection between combinatorial Hopf algebra structures and multiple zeta value identities.

Abstract

Recently, inspired by the Connes-Kreimer Hopf algebra of rooted trees, the second named author introduced rooted tree maps as a family of linear maps on the noncommutative polynomial algebra in two letters. These give a class of relations among multiple zeta values, which are known to be a subclass of the so-called linear part of the Kawashima relations. In this paper we show the opposite implication, that is the linear part of the Kawashima relations is implied by the relations coming from rooted tree maps.