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

Abstract

We study two linear bases of the free associative algebra Z⟨X, Y ⟩: one is formed by the Magnus polynomials of type (adand 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 Z⟨X, Y ⟩. As an application, we derive a formula of Le-Murakami, Furusho type that expresses arbitrary coefficients of a group-like series J ∈ C⟨⟨X, Y ⟩⟩ in terms of the "regular" coefficients of J. 0 of R⟨X, Y ⟩ (to be called the Magnus polynomials below) designed as follows: Manuscript