Decomposition of free Lie algebras into irreducible components (Q1376333)

Let \(G\) be a group, \(V\) a finite dimensional \(G\)-module and \({\mathcal L}(V)\) the free Lie algebra generated by \(V\). Then \({\mathcal L}(V)\) decomposes into homogeneous subspaces \({\mathcal L}_n(V)\) spanned by \([\cdots[[v_1v_2]v_3]\cdots v_n]\), where \(v_i\in V\). In the article by \textit{V. G. Kac} and \textit{S.-J. Kang} [``Representations of Groups'', Can. Math. Soc. Conf. Proc. 16, 141-154 (1995)], a trace formula for \({\mathcal L}_n(V)\) is given. In the article under review, the authors use this formula to compute the growth of multiplicities of irreducible components in \({\mathcal L}_n(V)\) as \(n\) varies. They also decompose \({\mathcal L}_{\alpha}(V_1\oplus\cdots \oplus V_k)\) and \({\mathcal L}_n(V_1\otimes\cdots\otimes V_k)\) into irreducible \(GL(V_1)\times\cdots\times GL(V_k)\)-modules.