TY - JOUR
T1 - Proof of a conjecture of Bergeron, Ceballos and Labbé
AU - Postnikov, Alexander
AU - Grinberg, Darij
N1 - Publisher Copyright:
© 2017, University at Albany. All rights reserved.
Copyright:
Copyright 2018 Elsevier B.V., All rights reserved.
PY - 2017/11/4
Y1 - 2017/11/4
N2 - 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 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.
AB - 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 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.
KW - Combinatorics
KW - Coxeter groups
KW - Group theory
KW - Reduced expressions
KW - Spin symmetric groups
KW - Symmetric groups
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M3 - Article
AN - SCOPUS:85039725245
SN - 1076-9803
VL - 23
SP - 1581
EP - 1610
JO - New York Journal of Mathematics
JF - New York Journal of Mathematics
ER -