Coaction formula for the motivic version of Yamamoto’s integral

Abstract We introduce a motivic version of Yamamoto’s integrals, which are integrals attached to labeled posets. For each labeled poset, this motivic version is defined as a sum of motivic iterated integrals. We extend both the infinitesimal coaction and the coaction from motivic iterated integrals to the motivic version of Yamamoto’s integrals. Our main theorem provides an explicit formula for the infinitesimal coaction on the motivic version of Yamamoto’s integrals. The summands in this formula are canonically indexed by irreducible labeled subposets, and, in the totally ordered case, the formula recovers the usual infinitesimal coaction formula for motivic iterated integrals. In addition, we prove an explicit summation formula for the contributions to the coaction expansion associated with a fixed subposet. This gives a method for computing the coaction, and, in the totally ordered case, this summation recovers the corresponding terms in the coaction formula for motivic iterated integrals. As an application, we study Schur multiple zeta values with constant diagonal entries, for which integral expressions are known, and use our formulas to compute the infinitesimal coaction and the coaction for certain classes of motivic Yamamoto integrals.