Factoring Euclidean isometries

Every isometry of a finite dimensional euclidean space is a product of reflections and the minimum length of a reflection factorization defines a metric on its full isometry group. In this article we identify the structure of intervals in this metric space by constructing, for each isometry, an explicit combinatorial model encoding all of its minimal length reflection factorizations. The model is largely independent of the isometry chosen in that it only depends on whether or not some point is fixed and the dimension of the space of directions that points are moved. Every good geometry book proves that each isometry of euclidean nspace is a product of at most n+1 reflections and several more-advanced sources include Scherk’s theorem which identifies the minimal length of such a reflection factorization from the basic geometric attributes of the isometry under consideration [Sch50, Die71, ST89, Tay92]. The structure of the full set of minimal length reflection factorizations, on the other hand, does not appear to have been given an elementary treatment in the literature even though the proof only requires basic geometric tools. In this article we construct, for each isometry, an explicit combinatorial model encoding all of its minimal length reflection factorizations. The model is largely independent of the isometry chosen in that it only depends on whether or not some point is fixed and the dimension of the space of directions that points are moved. Analogous results for spherical isometries already exist and are easy to state: when w is an orthogonal linear transformation of R only fixing the origin, for example, there is a natural bijection between minimal length factorizations of w into reflections fixing the origin and complete flags of linear subspaces in R [BW02]. In other words, the structure of all such factorizations is encoded in the lattice of linear subspaces of R with one factorization for each maximal chain. We construct a Date: December 27, 2013. 1The only discussions of this issue that we have found in the literature use Wall’s parameterization of the orthogonal group and the main results are stated in terms of inclusions of nondegenerate subspaces under an asymmetric bilinear form derived from the original isometry [Tay92, Wal63].