A characterization of root systems from the viewpoint of denominator formulae

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.