Proof of a conjecture of Bergeron, Ceballos and Labbé

• Main Authors: Postnikov, Alexander (Contributor), Grinberg, Darij (Contributor)
• Other Authors: Massachusetts Institute of Technology. Department of Mathematics (Contributor)
• Format: Article
• Language: English
• Published: State University of New York at Albany, 2018-06-12T16:29:26Z.
• Subjects: Article
• Online Access: Get fulltext

 © 2017, University at Albany. All rights reserved. The reduced expressions for a given element w of a Coxeter group (W, S) can be regarded as the vertices of a directed graph R(w); its arcs correspond to the braid moves. Specifically, an arc goes from a reduced expression a to a reduced expression b when b is obtained from a by replacing a contiguous subword of the form stst ... (for some distinct s,t ∈ S) by tsts ... (where both subwords have length m s,t , the order of st ∈ W). We prove a strong bipartiteness-type result for this graph R(w): Not only does every cycle of R(w) have even length; actually, the arcs of R(w) can be colored (with colors corresponding to the type of braid moves used), and to every color c corresponds an "opposite" color c op (corresponding to the reverses of the braid moves with color c), and for any color c, the number of arcs in any given cycle of R(w) having color in {c, c op } is even. This is a generalization and strengthening of a 2014 result by Bergeron, Ceballos and Labbé.