The motivic Galois group for a double zeta value

Abstract: We consider multiple zeta values, which are periods of mixed Tate motives over $\mathbb Z$. For a given multiple zeta value $ζ$, there exists a unique minimal motive $M(ζ)$ such that $ζ$ is a period of $M(ζ)$. In general, the motive $M(ζ)$ is difficult to compute. In this article, we compute the minimal motive $M(a,b)$ associated to a given double zeta value $ζ(a,b)$. We also compute the motivic Galois group $G(a,b)$ associated to $ζ(a,b)$ and discuss its dimension. Moreover, we give a period matrix of $M(a,b)$. The period conjecture predicts that the dimension of $G(a,b)$ equals the transcendence degree of the algebra of periods of $M(a,b)$. Hence our results lead to conjectures about algebraic relations between single and double zeta values.