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

Abstract

We study two linear bases of the free associative algebra $\mathbb{Z}⟨X,Y⟩$: 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 $\mathbb{Z}⟨X,Y⟩$. As an application, we show a formula of Le-Murakami, Furusho type that expresses arbitrary coefficients of a group-like series $J\in \mathbb{C}⟨⟨X,Y⟩⟩$ by the `regular' coefficients of $J$.

arXiv commentAlgebraic Combinatorics (to appear)