Groups satisfying Scherk's length theorem
Let \(V\) be a vector space over a field \(K\) of characteristic \(\neq 2\), and let \(G\) be a subgroup of the general linear group \(\text{GL}(n,K)\). Assume \(G\) is generated by its simple involutions such that \(-1_V\in G\) and Scherk's length theorem holds for \(G\) (i.e., \(G\) is a reflection group). The authors show that in any such irreducible group \(G\) there is a sequence of subspaces, each having an inner point, such that its sum is the entire space. They construct successively a quadratic form on sums of such spaces. Thus any group \(G\) having the above-mentioned properties is a subgroup of an orthogonal group.
scientific article; zbMATH DE number 1146246