Integral Lie powers of a lattice for the cyclic group of order 2 (Q1590716)

Let \(C_2\) be a cyclic group of order 2. There are three indecomposable \({\mathbb{Z}}C_2\)-lattices, up to isomorphism. Let \(L_n\) denote the \(n\)-th homogeneous component of the free Lie ring \(L(W)\) on a given \({\mathbb{Z}}C_2\)-lattice \(W\). This paper gives explicit formulae for the multiplicities of the three indecomposable \({\mathbb{Z}}C_2\)-lattices in a Krull-Schmidt decomposition of \(L_n\). As an application, the structure of the higher dimensional modules associated to a non-cyclic free presentation of \(C_2\) is determined.