A survey of total positivity

TL;DR

This survey reviews the development of total positivity, its extension to Lie groups, and recent advances in understanding surface group homomorphisms within this framework.

Contribution

It compiles and explains the evolution of total positivity concepts and their applications to Lie groups and surface group representations.

Findings

Reduction of bilinear forms to canonical forms using total positivity

Connection between total positivity and Lie group structures

Summary of Fock and Goncharov's results on surface group homomorphisms

Abstract

We survey the history of totally positive matrices and the generalization to Lie groups. We describe a reduction of a bilinear form to a canonical form (generalizing the case of symplectic nondegenerate forms) using ideas from total positivity; we also place this in a Lie group context. We give a short exposition of results of Fock and Goncharov on the study of homomorphisms of the fundamental group of a closed surface into a Lie group.