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

TL;DR

This paper explores the application of Lusztig's total positivity theory to Chevalley groups, explicitly characterizing regions of total positivity related to root subgroups within the root category.

Contribution

It explicitly determines regions of total positivity for Chevalley groups using root categories, extending the understanding of total positivity in this context.

Findings

Explicit regions of total positivity are identified for Chevalley groups.

The size of monoids of totally positive elements is described in terms of root subgroups.

The work connects root categories with total positivity in algebraic groups.

Abstract

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.