On Inversion Triples and Braid Moves
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Annals of Combinatorics
On Inversion Triples and Braid Moves Jozsef Losonczy Abstract. An inversion triple of an element w of a simply laced Coxeter group W is a set {α, β, α + β}, 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 {α, β, α + β}. This enables us to prove rather quickly that every triple of positive roots {α, β, α + β} 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. Mathematics Subject Classification. 05E15, 20F55. Keywords. Coxeter group, Braid move, Root sequence, Inversion triple, Contractible triple, Non-gatherable triple.
1. Introduction Let W be a simply laced Coxeter group with set S of distinguished generators. To each reduced expression for a group element w, one can assign a sequence, called a “root sequence” for w, whose terms are the positive roots sent negative by w. The assignment of root sequences to reduced expressions is a bijection, and, relative to this bijection, a braid move sts → tst applied to a reduced expression corresponds to a substitution (α, α + β, β) → (β, α + β, α) on the root sequence side. A commutation move applied to a reduced expression corresponds to a commutation of two roots in the corresponding root sequence. We 0123456789().: V,-vol
J. Losonczy
can, therefore, approach the combinatorics of reduced expressions by focusing on root sequences. A set of positive roots of the form {α, β, α + β} is called an “inversion triple” of an element w if the roots in the triple are all sent to negative roots by w. It is not difficult to show that if {α, β, α + β} is an inversion triple of w, then α + β must appear between α and β in any root sequence for w. An inversion triple of w is said to be “contractible” if the roots in the triple appear consecutively in some root sequence for w (thereby permitting a substitution as shown in the paragraph above). Not all inversion triples are contractible [13, Example 2.2.3]. We remark that, if the number of contractible inversion triples of w is denoted by N (w), then the number of commutation classes of reduced expressions for w is bounded above by 2N (w) [13, Sec
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