Total positivity for loop groups II: Chevalley generators

Abstract

This is the second in a series of papers developing a theory of total positivity for loop groups. In this paper, we study infinite products of Chevalley generators. We show that the combinatorics of infinite reduced words underlies the theory, and develop the formalism of infinite sequences of braid moves, called a braid limit. We relate this to a partial order, called the limit weak order, on infinite reduced words.The limit semigroup generated by Chevalley generators has a transfinite structure. We prove a form of unique factorization for its elements, in effect reducing their study to infinite products which have the order structure of ℕ. For the latter infinite products, we show that one always has a factorization which matches an infinite Coxeter element.One of the technical tools we employ is a totally positive exchange lemma which appears to be of independent interest. This result states that the exchange lemma (in the context of Coxeter groups) is compatible with total positivity in the form of certain inequalities.