Module structure of the free Lie ring on three generators

Let Lⁿ denote the homogeneous component of degree n in the free Lie ring on three generators, viewed as a module for the symmetric group S₃ of all permutations of those generators. This paper gives a Krull-Schmidt Theorem for the Lⁿ: if n > 1 and Lⁿ is written as a direct sum of indecomposable submodules, then the summands come from four isomorphism classes, and explicit formulas for the number of summands from each isomorphism class show that these multiplicities are independent of the decomposition chosen. A similar result for the free Lie ring on two generators was implicit in a recent paper of R. M. Bryant and the second author. That work, and its continuation on free Lie algebras of prime rank p over fields of characteristic p, provide the critical tools here. The proof also makes use of the identification of the isomorphism types of ℤ-free indecomposable ℤS₃-modules due to M. P. Lee. (There are, in all, ten such isomorphism types, and in general there is no Krull-Schmidt Theorem for their direct sums.).