$t$-adic symmetrization map on harmonic algebra

TL;DR

This paper introduces a $t$-adic generalization of the symmetrization map on harmonic algebra, extending previous work to include elements from the theory of 2-colored rooted trees and providing explicit calculations.

Contribution

It develops a $t$-adic extension of the symmetrization map on harmonic algebra and computes its action on elements related to 2-colored rooted trees.

Findings

Defined the $t$-adic symmetrization map $\, extstyleigwedge ext{ extasciitilde}\, ext{on harmonic algebra}$

Calculated $\, extstyleigwedge ext{ extasciitilde}\,w$ for elements from 2-colored rooted trees

Extended explicit formulas for symmetrization to a $t$-adic setting

Abstract

Bachmann, Takeyama and Tasaka introduced a $\mathbb{Q}$-linear map $\phi$, which we call the symmetrization map in this paper, on the harmonic algebra $\mathfrak{H}^1$. They calculated $\phi(w)$ explicitly for an element $w$ in $\mathfrak{H}^1$ related to the multiple zeta values of Mordell--Tornheim type. In this paper, we introduce its $t$-adic generalization $\widehat{\phi}$ and calculate $\widehat{\phi}(w)$ for an element $w$ in $\mathfrak{H}^1[[t]]$ constructed from the theory of $2$-colored rooted tree.