Double shuffle relation for associators

Abstract

It is proved that Drinfel$’$d’s pentagon equation implies the generalized double shuffle relation. As a corollary, an embedding from the Grothendieck-Teichmüller group ${\rm GRT}_1$ into Racinet’s double shuffle group ${\rm DMR}_0$ is obtained, which settles the project of Deligne-Terasoma. It is also proved that the gamma factorization formula follows from the generalized double shuffle relation.