Factoring Euclidean isometries
Noel Brady, Jon McCammond · International Journal of Algebra and Computation · 2015
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 paper we identify the structure of intervals in this metric space by constructing, for each isometry, an explicit combinatorial model encoding all of its minimum 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.