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| Format: | Preprint |
| Veröffentlicht: |
2016
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| Online-Zugang: | https://arxiv.org/abs/1603.03138 |
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| _version_ | 1866910123401674752 |
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| author | Grinberg, Darij Postnikov, Alexander |
| author_facet | Grinberg, Darij Postnikov, Alexander |
| contents | The reduced expressions for a given element $w$ of a Coxeter group $(W, S)$ can be regarded as the vertices of a directed graph $\mathcal{R}(w)$; 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 $s, t$ in $S$) by $tsts...$ (where both subwords have length $m_{s, t}$, the order of $st$ in $W$). We prove a strong bipartiteness-type result for this graph $\mathcal{R}(w)$: Not only does every cycle of $\mathcal{R}(w)$ have even length; actually, the arcs of $\mathcal{R}(w)$ can be colored (with colors corresponding to the type of braid moves used), and to every color $c$ corresponds an "opposite" color $c^{\operatorname{op}}$ (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 $\mathcal{R}(w)$ having color in $\left\{c, c^{\operatorname{op}}\right\}$ is even. This is a generalization and strengthening of a 2014 result by Bergeron, Ceballos and Labbé. We state further conjectural extensions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1603_03138 |
| institution | arXiv |
| publishDate | 2016 |
| record_format | arxiv |
| spellingShingle | Proof of a conjecture of Bergeron, Ceballos and Labbé Grinberg, Darij Postnikov, Alexander Combinatorics 20F55 The reduced expressions for a given element $w$ of a Coxeter group $(W, S)$ can be regarded as the vertices of a directed graph $\mathcal{R}(w)$; 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 $s, t$ in $S$) by $tsts...$ (where both subwords have length $m_{s, t}$, the order of $st$ in $W$). We prove a strong bipartiteness-type result for this graph $\mathcal{R}(w)$: Not only does every cycle of $\mathcal{R}(w)$ have even length; actually, the arcs of $\mathcal{R}(w)$ can be colored (with colors corresponding to the type of braid moves used), and to every color $c$ corresponds an "opposite" color $c^{\operatorname{op}}$ (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 $\mathcal{R}(w)$ having color in $\left\{c, c^{\operatorname{op}}\right\}$ is even. This is a generalization and strengthening of a 2014 result by Bergeron, Ceballos and Labbé. We state further conjectural extensions. |
| title | Proof of a conjecture of Bergeron, Ceballos and Labbé |
| topic | Combinatorics 20F55 |
| url | https://arxiv.org/abs/1603.03138 |