将二次空间的等距分解为反射
Factoring isometries of quadratic spaces into reflections
摘要：
设$V$是具有非退化二次型$Q$的向量空间。如果基本场$\mathbb{F}$与$\mathbb{F}_2$不同，则已知每个等距都可以被写为反射的乘积。在这篇文章中，我们详细的结构的偏序集的所有最小长度反射因子分解的等距。如果$\mathbb{F}$是一个序域，我们还研究了正反射的因子分解，即，由正范数向量定义的反射。我们的特点，这样的因式分解，在假设下，$\mathbb{F}$的平方是密集的正元素（这包括阿基米德和欧几里得领域）。特别地，我们证明了等距是正反射的乘积当且仅当它的旋量范是正的。作为最后的应用，我们明确地描述了双曲空间的等距的所有因子分解的偏序集。
英文摘要：
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.
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