Module structure of the free Lie ring on three generators (Q1125389)

Let \(L\) be a free Lie ring on three generators and let \(L^{n}\) denote the homogeneous component of degree \(n\) viewed as a module for the symmetric group \(S_{3}\) whose elements permute the generators. The authors show that \(L^{n}\) is the direct sum of indecomposable submodules of four types and explicit formulas for the number of summands of each of these four types are given.