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This paper concerns the associative lower central series ideals $M_i$ of thefree algebra $A_n$ on $n$ generators. Namely, we study the successive quotients$N_i=M_i/M_{i+1}$, which admit an action of the Lie algebra $W_n$ of vectorfields on $\Bbb C^n$. We bound the degree $|\lambda|$ of tensor field modules$F_\lambda$ appearing in the Jordan-H\"older series of each $N_i$, confirming arecent conjecture of Arbesfeld and Jordan. As an application, we compute thesedecompositions for small $n$ and $i$.

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