On the filtration of a free algebra by its associative lower central series

This paper concerns the associative lower central series ideals $M_i$ of the free algebra $A_n$ on $n$ generators. Namely, we study the successive quotients $N_i=M_i\mathrm{/}{M}_{i+1}$, which admit an action of the Lie algebra $W_n$ of vector fields on ${\mathbb{C}}^n$. We bound the degree $\mathrm{\mid }\lambda \mathrm{\mid }$ of tensor field modules $F_{\lambda }$ appearing in the Jordan-H"older series of each $N_i$, confirming a recent conjecture of Arbesfeld and Jordan. As an application, we compute these decompositions for small $n$ and $i$.