From Weyl groups to semisimple groups

TL;DR

This paper explores how properties of complex semisimple groups can be characterized using Weyl groups and introduces new links between canonical bases and total positivity.

Contribution

It presents a novel approach to describing semisimple groups via Weyl groups and uncovers new connections between canonical bases and total positivity.

Findings

Properties of semisimple groups can be expressed through Weyl groups

New relationships between canonical bases and total positivity are established

The approach simplifies understanding of complex semisimple groups

Abstract

In this (partly expository) paper we show, using ideas from the theory of total positivity, how a number of properties of a semisimple group over the complex numbers can be presented purely in terms of the Weyl group. We also describe some new connections of the theory of canonical bases with total positivity.