Lie powers and Witt vectors
Let \(k\) be a field of characteristic \(p>0\) and \(G\) a group. Suppose that \(G\) has a finite dimensional representation in a space \(V\). Denote by \(L(V)\) the free Lie algebra on \(V\). Then each homogeneous component \(L^n(V)\) is a finite dimensional representation of \(G\). Let \(m\) be a positive integer which is not divisible by \(p\). As it is shown by the author and \textit{M. Schocker} in the paper [Proc. Lond. Math. Soc. (3) 93, No. 1, 175--196 (2006; Zbl 1174.17006)], there exists a \(kG\)-module \(B_{p^lm} \subseteq L^{p^lm}\) which is a direct summand of \(V^{\otimes p^lm}\) such that \[ L^{p^lm} = L^{p^l}\left(B_m\right) \oplus L^{p^l-1}\left(B_{pm}\right) \oplus \cdots \oplus L^1\left(B_{p^lm}\right). \] In the present paper it is shown that the module \(V^{p^lm}\) has a decomposition \(\bigoplus _{d\mid m}\phi(d) U_{m,d}\) for some specific modules \(U_{m,d}\) and \[ \left(V^{\otimes m}\right)^{\otimes p^lm} = \bigoplus_{\lambda \in\Lambda} U_{m,\lambda(1)}\otimes \cdots \otimes U_{m,\lambda(p^l)} \] where \(\Lambda\) is a finite set of \(p^l\)-tuples \(\left(\lambda(1), \ldots, \lambda(p^l)\right)\) with entries \(\lambda(1),\ldots \) which are divisors of \(m\). Moreover \(B_{p^lm}\) has a decomposition \[ B_{p^lm} = \bigoplus_{\lambda\in \Lambda_0} U_{m,\lambda(1)}\otimes \cdots \otimes U_{m,\lambda(p^l)} \] for some specific subset \(\Lambda_0\) in \(\Lambda\).
scientific article; zbMATH DE number 5566028
scientific article