Summary: We give two combinatorial properties of elements satisfying the stuffle relations; one showing that double shuffle elements are determined by less than the full set of stuffle relations, and the other a cyclic property of their coefficients. Although simple, the properties have some useful applications, of which we give two. The first is a generalization of a theorem of Ihara on the abelianizations of elements of the Grothendieck-Teichmüller Lie algebra \(\mathfrak{grt}\) to elements of the double shuffle Lie algebra in a much larger quotient of the polynomial algebra than the abelianization, namely the trace quotient introduced by A. Alekseev and C. Torossian [Ann. Math. (2) 175, No. 2, 415–463 (2012; Zbl 1243.22009)]. The second application is a proof that the Grothendieck-Teichmüller Lie algebra \(\mathfrak{grt}\) injects into the double shuffle Lie algebra \(\mathfrak{ds}\), based on the recent proof by H. Furusho [Ann. Math. (2) 174, No. 1, 341–360 (2011; Zbl 1321.11088)] of this theorem in the pro-unipotent situation, but in which the combinatorial properties provide a significant simplification.
For the entire collection see [Zbl 1257.00018].