Proof of a conjecture of Bergeron, Ceballos and Labbé
© 2017, University at Albany. All rights reserved. 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...
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| Format: | Article |
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State University of New York at Albany
2018
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| Online Access: | http://hdl.handle.net/1721.1/116265 https://orcid.org/0000-0002-3964-8870 https://orcid.org/0000-0002-9661-8432 |
| _version_ | 1811086501492031488 |
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| author | Postnikov, Alexander Grinberg, Darij |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Postnikov, Alexander Grinberg, Darij |
| author_sort | Postnikov, Alexander |
| collection | MIT |
| description | © 2017, University at Albany. All rights reserved. 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 m s,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 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. |
| first_indexed | 2024-09-23T13:26:56Z |
| format | Article |
| id | mit-1721.1/116265 |
| institution | Massachusetts Institute of Technology |
| last_indexed | 2024-09-23T13:26:56Z |
| publishDate | 2018 |
| publisher | State University of New York at Albany |
| record_format | dspace |
| spelling | mit-1721.1/1162652022-10-01T15:23:43Z Proof of a conjecture of Bergeron, Ceballos and Labbé Postnikov, Alexander Grinberg, Darij Massachusetts Institute of Technology. Department of Mathematics Postnikov, Alexander Grinberg, Darij © 2017, University at Albany. All rights reserved. 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 m s,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 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. 2018-06-12T16:29:26Z 2018-06-12T16:29:26Z 2017-10 2018-05-29T18:13:54Z Article http://purl.org/eprint/type/JournalArticle 1076-9803 http://hdl.handle.net/1721.1/116265 Postnikov, Alexander and Darij Grinberg. "Proof of a conjecture of Bergeron, Ceballos and Labbé." New York Journal of Mathematics 23 (2017), pp. 1581-1610. https://orcid.org/0000-0002-3964-8870 https://orcid.org/0000-0002-9661-8432 http://nyjm.albany.edu/j/2017/23-70.html New York Journal of Mathematics Creative Commons Attribution-Noncommercial-Share Alike http://creativecommons.org/licenses/by-nc-sa/4.0/ application/pdf State University of New York at Albany arXiv |
| spellingShingle | Postnikov, Alexander Grinberg, Darij Proof of a conjecture of Bergeron, Ceballos and Labbé |
| title | Proof of a conjecture of Bergeron, Ceballos and Labbé |
| title_full | Proof of a conjecture of Bergeron, Ceballos and Labbé |
| title_fullStr | Proof of a conjecture of Bergeron, Ceballos and Labbé |
| title_full_unstemmed | Proof of a conjecture of Bergeron, Ceballos and Labbé |
| title_short | Proof of a conjecture of Bergeron, Ceballos and Labbé |
| title_sort | proof of a conjecture of bergeron ceballos and labbe |
| url | http://hdl.handle.net/1721.1/116265 https://orcid.org/0000-0002-3964-8870 https://orcid.org/0000-0002-9661-8432 |
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