Notes on Chevalley Groups and Root Category III: the Region of Total Positivity

In [4], we use the root categories to realize Chevalley groups. Lusztig’s theory of total positivity for reductive groups can be naturally applied to Chevalley groups. In this paper, we explicitly determine regions of ${\mathbb{R}}_{>0}^t$ for describing the size of monoids of totally positive elements, with respect to the root subgroups corresponding to the indecomposable objects in the root category.