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Let $\sigma$ be a permutation. If two positive braids represent $\sigma$ and are of minimal length among the braids representing $\sigma$, then they are equal. From what I could gather, this result is Theorem 9.2.5 in:

Epstein, David, et al. Word processing in groups. AK Peters, Ltd., 1992.

This book refers to a paper by Garside called "The Braid group and other groups", but I can't seem to find the result in Garside's paper.

I was told that this result can be generalised, replacing the positive braids by elements of an Artin monoid, and the permutation group by the associated Coxeter group. Where can this result be found?

I was told that this result appears in "Characters of Finite Coxeter Groups and Iwahori-Hecke Algebras", but I could not find it, probably because I am not familiar enough with the theory to recognise the result that I am looking for.

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  • $\begingroup$ It doesn't sound right. What about $a^2$ and $b^2$ for two generators $a$ and $b$ of the braid group? $\endgroup$ – Derek Holt Nov 30 '16 at 11:16
  • $\begingroup$ @DerekHolt Oops You are right! I meant to say that the braids have to be of minimal length. I'll edit my question. Thank you! $\endgroup$ – Maxime Lucas Nov 30 '16 at 11:18
  • $\begingroup$ Does ems-ph.org/journals/… answer what you want? $\endgroup$ – Benjamin Steinberg Nov 30 '16 at 11:29
  • $\begingroup$ @BenjaminSteinberg Unless I am missing something I don't think so. The article you are linking to refers to the inclusion Artin monoid -> Artin groups (that is, positive braids into braids), but does not talk about Coxeter groups. $\endgroup$ – Maxime Lucas Nov 30 '16 at 11:35
  • $\begingroup$ Sorry I misread the question. $\endgroup$ – Benjamin Steinberg Nov 30 '16 at 12:09
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This is a general fact on Artin-Tits groups attached to Coxeter groups. Let $B(W)^+$ be the Artin-Tits monoid of the Coxeter group $W$. Then there is a canonical surjection $B(W)^+\rightarrow W$ which you mentioned.

Now starting from any reduced expression $s_1 s_2 \cdots s_k$ of an element $w$ in $W$ where $s_i$ are elements of the generating set $S$, replace each $s_i$ in this reduced expression by the corresponding generator $\sigma_i$ of the Artin monoid to get a word $\sigma_1 \sigma_2\cdots \sigma_k$. It turns out that the element $b\in B(W)^+$ represented by this word is independent of the chosen reduced expression for $w$, as a consequence of the so-called Matsumoto Lemma in Coxeter groups ("Générateurs et relations des groupes de Weyl généralisés", C. R. Acad. Sci. Paris, 258. 3419–3422): it states that one can pass from any reduced expression of $w$ to any other just by applying a sequence of braid relations (i.e. without needing to increase the length of the reduced word), which are the relations defining $B(W)^+$, hence also hold in $B(W)^+$. This gives a set-theoretic section $W \rightarrow B(W)^+$ of the surjection above. Mastumoto's Lemma works for arbitrary Coxeter groups and you will find it in most books on Coxeter groups.

As a consequence one gets your claim.

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