 

Alexander Postnikov and Darij Grinberg
Proof of a conjecture of Bergeron, Ceballos and Labbé view print


Published: 
November 4, 2017

Keywords: 
Coxeter groups, reduced expressions, symmetric groups, spin symmetric groups, combinatorics, group theory 
Subject: 
20F55 


Abstract
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 m_{s,t}, the order of st∈ W). We prove a strong
bipartitenesstype 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.


Author information
Alexander Postnikov:
Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, United States
apost@math.mit.edu
Darij Grinberg:
Dunham Jackson Assistant Professor, School of Mathematics, University of Minnesota, 127 Vincent Hall, 206 Church St. SE, Minneapolis, MN 55455, USA
dgrinber@umn.edu

