New York Journal of Mathematics
Volume 23 (2017) 1581-1610


Alexander Postnikov and Darij Grinberg

Proof of a conjecture of Bergeron, Ceballos and Labbé

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Published: November 4, 2017
Keywords: Coxeter groups, reduced expressions, symmetric groups, spin symmetric groups, combinatorics, group theory
Subject: 20F55

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 \overrightarrow{a} to a reduced expression \overrightarrow{b} when \overrightarrow{b} is obtained from \overrightarrow{a} by replacing a contiguous subword of the form stst... (for some distinct s,t∈ S) by tsts... (where both subwords have length ms,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 cop (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,cop} is even. This is a generalization and strengthening of a 2014 result by Bergeron, Ceballos and Labbé.

Author information

Alexander Postnikov:
Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, United States

Darij Grinberg:
Dunham Jackson Assistant Professor, School of Mathematics, University of Minnesota, 127 Vincent Hall, 206 Church St. SE, Minneapolis, MN 55455, USA