Lie powers and pseudo-idempotents (Q2882476)

For a vector space \(V\) over a field \(K\) let \(T=T(V)\) be its tensor algebra, \(L(V) \subset T(V)\) the free Lie algebra generated by \(V\), and \(L_n(V)=T_n(V)\cap L(V)\), where \(T_n(V)=V^{\otimes n}\).NEWLINENEWLINEBoth \(T_n(V)\) and \(L_n(V)\) are \(\mathrm{GL}(V)\)-modules and it is known that \(L_n(V)\) is a direct summand in \(T_n(V)\) provided \(\text{char}(K)\neq n\). Under the same assumption on \(K\), the authors construct an explicit projection showing that \(L_2(L_n(V))\) is a direct summand of \(L_{2n}(V)\) as a \(\mathrm{GL}(V)\)-module, a particular case of results on decomposition of \(L_n(V)\) as a \(\mathrm{GL}(V)\)-module obtained in [\textit{R. M. Bryant} and \textit{M. Schocker}, Proc. Lond. Math. Soc. (3) 93, No. 1, 175--196 (2006; Zbl 1174.17006)].NEWLINENEWLINETo this end, they give an interesting new factorization of the Dynkin operator \(\Omega_n\), an element of the group ring \(\mathbb{Z}S_n\) satisfying \(\Omega_n^2=n\Omega_n\) (what the authors call a pseudo-idempotent) which, provided \(\text{char}(K)\neq n\), gives rise to the projection from \(T_n(V)\) to \(L_n(V)\) as mentioned above.