Demi-shuffle duals of Magnus polynomials in a free associative algebra

TL;DR

This paper introduces a new dual basis called demi-shuffle polynomials in the free associative algebra, providing a formula to relate coefficients of group-like series to regular coefficients, advancing algebraic understanding.

Contribution

It constructs and analyzes the demi-shuffle dual basis to Magnus polynomials in a free algebra, and derives a formula connecting series coefficients.

Findings

Demi-shuffle basis is dual to Magnus polynomials.

Derived a formula relating coefficients of group-like series.

Enhanced algebraic tools for free associative algebras.

Abstract

We study two linear bases of the free associative algebra $\mathbb{Z}\langle X,Y\rangle$: one is formed by the Magnus polynomials of type $(\mathrm{ad}_X^{k_1}Y)\cdots(\mathrm{ad}_X^{k_d}Y) X^k$ and the other is its dual basis (formed by what we call the `demi-shuffle' polynomials) with respect to the standard pairing on the monomials of $\mathbb{Z}\langle X,Y\rangle$. As an application, we show a formula of Le-Murakami, Furusho type that expresses arbitrary coefficients of a group-like series $J\in \mathbb{C}\langle\langle X,Y\rangle\rangle$ by the `regular' coefficients of $J$.