TY - JOUR

T1 - Proof of a conjecture of Bergeron, Ceballos and Labbé

AU - Postnikov, Alexander

AU - Grinberg, Darij

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

VL - 23

SP - 1581

EP - 1610

JO - New York Journal of Mathematics

JF - New York Journal of Mathematics

SN - 1076-9803

ER -