标题： 从分母公式的角度刻画根系 显示英文标题

标题： A characterization of root systems from the viewpoint of denominator formulae

摘要： 根系是一些具有卓越对称性的集合，因此它们出现在数学的许多场合中。 其中，根系的分母公式是非常优美且神秘的方程，从数学的各个学科来看，它具有多重意义。 本文展示了这一现象的一个逆命题。 也就是说，对于欧几里得向量空间 $ S $ 的一个有限子集 $ V $，在群环 $ F $ 中定义一个类似于分母公式乘积部分的方程 $ {\mathbb{Z}}[V] $。 然后， $ F $ 支持的几何条件刻画了 $ S $ 是否为有限/仿射根系正根集的集合，从而恢复分母公式。 这为约化有限/仿射根系正根集提供了新的特征描述。 显示英文摘要

摘要： Root systems are sets with remarkable symmetries and therefore they appear in many situations in mathematics. Among others, denominator formulae of root systems are very beautiful and mysterious equations which have several meanings from a variety of disciplines in mathematics. In this paper, we show a converse statement of this phenomena. Namely, for a given finite subset $ S $ of a Euclidean vector space $ V $, define an equation $ F $ in the group ring $ {\mathbb{Z}}[V] $ featuring the product part of denominator formulae. Then, a geometric condition for the support of $ F $ characterizes $ S $ being a set of positive roots of a finite/affine root system, recovering the denominator formula. This gives a novel characterization of the sets of positive roots of reduced finite/affine root systems.