Factoring isometries of quadratic spaces into reflections

TL;DR

This paper explores how isometries of quadratic spaces can be factored into reflections, detailing the structure of minimal factorizations, and extends the analysis to positive reflections over ordered fields, with applications to hyperbolic space.

Contribution

It provides a detailed description of the poset of minimal reflection factorizations and characterizes factorizations into positive reflections over ordered fields.

Findings

Every isometry can be expressed as a product of reflections (excluding characteristic 2).

Characterization of positive reflection factorizations via the spinor norm.

Explicit description of factorizations in hyperbolic space.

Abstract

Let $V$ be a vector space endowed with a non-degenerate quadratic form $Q$. If the base field $\mathbb{F}$ is different from $\mathbb{F}_2$, it is known that every isometry can be written as a product of reflections. In this article, we detail the structure of the poset of all minimal length reflection factorizations of an isometry. If $\mathbb{F}$ is an ordered field, we also study factorizations into positive reflections, i.e., reflections defined by vectors of positive norm. We characterize such factorizations, under the hypothesis that the squares of $\mathbb{F}$ are dense in the positive elements (this includes Archimedean and Euclidean fields). In particular, we show that an isometry is a product of positive reflections if and only if its spinor norm is positive. As a final application, we explicitly describe the poset of all factorizations of isometries of the hyperbolic space.