On multicommutators for simple algebraic groups (Q5958865)

Let \(G\) be a group and \([G,G]\) its commutator subgroup. The authors are interested in the following question: Given an \(n\)-tuple of commutators \([a_i,b_i]\), when is it possible to write \([a_i,b_i]=[g,g_i]\) using the same \(g\in G\) for all \(i\)?NEWLINENEWLINENEWLINELet \(\Phi_n\) be a mapping of \(G\times G^n\) into \(G^n\) such that \(\Phi_n(g,g_1,\dots,g_n)=([g,g_1],\dots,[g,g_n])\). If \(\text{Im }\Phi_n=[G,G]^n\), then \(G\) is said to have property \({\mathcal C}_n\). A simple algebraic group is said to have property \(\overline{\mathcal C}_n\) if the map \(\Phi_n\) is dominant, i.e., if \(G^n\) is the Zariski closure of \(\text{Im }\Phi_n\) in \(G^n\).NEWLINENEWLINENEWLINEThe main result in this paper is the following theorem: Let \(G\) be a simple algebraic group and let \(h=h(G)\) be the Coxeter number of the corresponding root system. Then \(G\) has the property \(\overline{\mathcal C}_n\) if and only if \(n\leq h+1\).NEWLINENEWLINENEWLINEThe authors prove a similar result for Lie algebras.