Factoring euclidean isometries

TL;DR

This paper explores the structure of isometries in finite-dimensional Euclidean spaces by constructing combinatorial models that encode all minimal reflection factorizations, revealing insights into their geometric and algebraic properties.

Contribution

It introduces an explicit combinatorial model for minimal reflection factorizations of Euclidean isometries, independent of specific isometries, based on fixed points and moved directions.

Findings

Constructed a combinatorial model for minimal reflection factorizations.

Identified the model's independence from specific isometries.

Provided a new perspective on the metric structure of Euclidean isometry groups.

Abstract

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.