Lie powers of free modules for certain groups of prime power order (Q5890417)

For a group \(G\), a field \(K\), and a finite-dimensional \(KG\)-module \(V\), the free Lie algebra \(L(V)\) on the \(K\)-space \(V\) is naturally a \(KG\)-module. For each positive integer \(n\), its homogeneous components \(L^n(V)\) is a submodule, called the \(n\)th Lie power of \(V\). Much information is available on the \(KG\)-module structure of \(L^n(V)\) if \(K\) has characteristic zero. In particular, if \(G\) is either finite or the general linear group on \(V\), then \(L^n(V)\) is semisimple, and results obtained in the 1940s by A.~Brandt and F.~Wever allow one to compute the multiplicities of the irreducible constituents of \(L^n(V)\). Much less is known if \(K\) has positive characteristic \(p\), where \(L^n(V)\) is usually not semisimple. Generally speaking, most difficulties appear to arise in case \(n\) is a multiple of \(p\).NEWLINENEWLINEThe \(KG\)-module structure of \(L^n(V)\) in case \(G\) is cyclic of order \(p\) and \(V\) is a free \(KG\)-module was determined by the first author and \textit{R. Stöhr} in [Trans. Am. Math. Soc. 352, 901--934 (2000; Zbl 1017.17007)]. In the paper under review that result is extended to the case where \(G\) is an arbitrary cyclic \(p\)-group or (when \(p=2\)) a quaternion or generalised quaternion group. Only two isomorphism types of indecomposable \(KG\)-modules occur as direct summands of \(L^n(V)\), namely the regular \(KG\)-module and the module induced from the indecomposable module of dimension \(p-1\) for the unique subgroup of \(G\) of order \(p\). Formulas for the multiplicities are given.