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Are there some references which proves the following result?

Let $W$ be a Coxeter group and $w \in W$. Then different reduced expressions of $w$ can be transformed from one into anther using only the braid relations.

Thank you very much.

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This is also known as Matsumoto's theorem. It was also independently proved by Jacques Tits.

There are tons of books having proofs of this, e.g. Theorem 3.3.1 in "Combinatorics of Coxeter" groups by Björner and Brenti. I am sure it's also in "Reflection Groups and Coxeter Groups" by Humphreys, but I don't have it at hand to give a precise reference.

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    $\begingroup$ Actually, I didn't deal with the combinatorial questions so explicitly in my book though I did discuss briefly the approach by Tits ams.org/mathscinet-getitem?mr=0254129 to the word problem in 8.1 (following a short remark on a less attractive method at the end of Chapter 5). While Matsumoto built on early work (of Iwahori. Tits) dealing with generators and relations, he didn't address the word problem for a Coxeter group as such. At any rate, the best exposition is in section 3.3-3.4 of the Bjorner-Brenti text. $\endgroup$ – Jim Humphreys Mar 16 '17 at 17:45

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