Proof of a conjecture of Bergeron, Ceballos and Labbé
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1603.03138.pdf
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307.15 KB
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6793647b85c90fceab90f5f16920ce67
Author(s)
Postnikov, Alexander
Grinberg, Darij
Date Issued
October 2017
Journal
New York Journal of Mathematics
Publisher
State University of New York at Albany
Citation
Postnikov, Alexander and Darij Grinberg. "Proof of a conjecture of Bergeron, Ceballos and Labbé." New York Journal of Mathematics 23 (2017), pp. 1581-1610.
Version
Author's final manuscript
Abstract
© 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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
http://nyjm.albany.edu/j/2017/23-70.html
