Proof of a conjecture of Bergeron, Ceballos and Labbé

Alexander Postnikov, Darij Grinberg

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

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 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.

Original languageEnglish (US)
Pages (from-to)1581-1610
Number of pages30
JournalNew York Journal of Mathematics
Volume23
StatePublished - Nov 4 2017

Keywords

  • Combinatorics
  • Coxeter groups
  • Group theory
  • Reduced expressions
  • Spin symmetric groups
  • Symmetric groups

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