Geometry of root elements in groups of type \(E_6\). (Q2892186)

Let \(G=G_{\mathrm{SC}}(\mathrm E_6,K)\) be the simply connected Chevalley group of type \(\mathrm E_6\) over a field \(K\), and let \(V\) be a minimal 27-dimensional module for \(G\). There exists a trilinear form \(F\colon V\times V\times V\to K\) such that \(G\) is the group of isometries of \(F\), and we define a subspace \(U\) of \(V\) to be \textit{singular} if \(F(u_1,u_2,v)=0\) for all \(u_1,u_2\in U\) and \(v\in V\) (in fact this definition must be slightly adjusted if \(\mathrm{char}(K)=2\)).NEWLINENEWLINE The aim of this paper is to study the relationship between the geometry of \(V\) and the root elements of \(G\). The primary result of the paper establishes a natural correspondence between root subgroups and six-dimensional singular subspaces. Indeed one can easily construct an injective map from the set of all root subgroups to the set of six-dimensional singular subspaces as follows: for a nontrivial root element \(g\in G\), define \(V^g:=\mathrm{Im}(g-I)\) where \(I\) is the identity. One can check that \(V^g\) is 6-dimensional and singular and, moreover, that the same singular subspace corresponds to elements of the same root subgroup, while different subspaces correspond to elements of different root subgroups. It remains to prove surjectivity, which is a difficult task.NEWLINENEWLINE Once this correspondence is proved, the author makes use of it to prove a result concerning the `relative position' of two root subgroups, the study of which has received a lot of attention in the literature. In particular the author connects the angle between two root elements \(g\) and \(h\) with the dimension of the singular subspace \(V^g\cap V^h\).NEWLINENEWLINE In the final section of the paper the author considers the problem of describing the subgroup generated by a triple of root subgroups, two of which are opposite. A full solution to this problem is given for the groups \(\mathrm{SO}_{2n}(K)\) and \(G_{\mathrm{SC}}(\mathrm E_6,K)\).NEWLINENEWLINE The author goes to some effort to motivate each of the main results with reference to the state of the literature. In addition the paper contains a wealth of background material concerning the group \(G_{\mathrm{SC}}(\mathrm E_6,K)\).