Superalgebras and dimensions of algebras

Several authors have used the representation theory of symmetric groups and superalgebras to prove that certain classes of algebras are nilpotent. We show how to extend these techniques to facilitate the computation of the dimensions of relatively free algebras, and we prove two general theorems which formalize the techniques. The ideas described here have been used in the computation of the dimensions of the associated Lie rings of free Engel-4 groups of exponent 5.