On Inversion Triples and Braid Moves

Abstract

An inversion triple of an element w of a simply laced Coxeter group W is a set $$\{ \alpha , \beta , \alpha + \beta \}$$ , where each element is a positive root sent negative by w. We say that an inversion triple of w is contractible if there is a root sequence for w in which the roots of the triple appear consecutively. Such triples arise in the study of the commutation classes of reduced expressions of elements of W, and have been used to define or characterize certain classes of elements of W, e.g., the fully commutative elements and the freely braided elements. Also, the study of inversion triples is connected with the representation theory of affine Hecke algebras and double affine Hecke algebras. In this paper, we describe the inversion triples that are contractible, and we give a new, simple characterization of the groups W with the property that all inversion triples are contractible. We also study the natural action of W on the set of all triples of (not necessarily positive) roots of the form $$\{ \alpha , \beta , \alpha + \beta \}$$ . This enables us to prove rather quickly that every triple of positive roots $$\{ \alpha , \beta , \alpha + \beta \}$$ is contractible for some w in W and, moreover, when W is finite, w may be taken to be the longest element of W. At the end of the paper, we pose a problem concerning the aforementioned action.