Factoring isometries of quadratic spaces into reflections

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.