Actions of automorphism groups of free groups on spaces of Jacobi diagrams. I

We consider an action of the automorphism group $\mathrm{Aut}(F_n)$ of the free group $F_n$ of rank $n$ on the filtered vector space $A_d(n)$ of Jacobi diagrams of degree $d$ on $n$ oriented arcs. This action induces on the associated graded vector space of $A_d(n)$, which is identified with the spa