Total positivity in loop groups: Chevalley generators

We study infinite products of Chevalley generators in the formal loop group. I will describe the formalism of infinite sequences of braid moves, called a braid limit. I will explain what are the equivalence classes of infinite reduced words, and how a partial order on them called the limit weak order arises. One interesting technical tool to be mentioned is the totally positive exchange lemma. It states that the usual exchange lemma in the context of Coxeter groups is compatible with total positivity in the form of certain inequalities. The talk is based on joint work with Thomas Lam.