Journal article
Proof of a conjecture of Bergeron, Ceballos and Labbe
New York journal of mathematics, Vol.23, pp.1581-1610
04 Nov 2017
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) over right arrow to a reduced expression (b) over right arrow when (b) over right arrow is obtained from (a) over right arrow by replacing a contiguous subword of the form stst. . . (for some distinct s, t is an element of S) by tsts. . . (where both subwords have length m(s,t), the order of st is an element of 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 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 Labbe.
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Details
- Title
- Proof of a conjecture of Bergeron, Ceballos and Labbe
- Creators
- Alexander Postnikov - MIT, Dept Math, 77 Massachusetts Ave, Cambridge, MA 02139 USADarij Grinberg - University of Minnesota
- Publication Details
- New York journal of mathematics, Vol.23, pp.1581-1610
- Publisher
- Electronic Journals Project
- Number of pages
- 30
- Resource Type
- Journal article
- Language
- English
- Academic Unit
- Mathematics
- Identifiers
- 991021862367004721
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