On the Weak Order of Orthogonal Groups

A complete lattice structure is defined on the underlying set of the orthogonal group of a real Euclidean space, by a construction analogous to that of the weak order of Coxeter systems in terms of root systems. This produces a complete rootoid in the sense of Dyer, with the orthogonal group as underlying group. It is shown that this complete lattice has a saturation property which is used along with other properties of the lattice to characterize the maximal totally ordered subsets of the lattice as collections of initial sections with respect to a total ordering on the positive roots.