TITLE: Algorithm for reducing words in a Coxeter group
QUESTION [8 upvotes]: Let $W$ be a Coxeter group with set of simple reflections $S$. Suppose that I have chosen a preferred reduced decomposition for every element of $W$. Given an arbitrary word in the alphabet $S$, is there an algorithm for reducing this word to my chosen decomposition using the Coxeter relations? That is to say, an algorithm which at each step replaces a subword of the form $s_i s_k s_i \dots$ by $s_k s_i s_k \dots$, or which removes an occurrence of $s_k^2$.
I would be happy with an answer in the following situation: $W=\Sigma_n$ is the symmetric group on $n$ letters and the preferred decomposition is given by taking the lexicographically smallest (or largest) reduced decomposition.

REPLY [11 votes]: An algorithm for reducing arbitrary words in any Coxeter group to their lexicographically least reduced representatives is described in the paper:
B. Brink and R.B. Howlett,
``A finiteness property and an automatic structure for Coxeter groups'',
Math. Ann. 296, (1993), 179-190.
It doesn't do exactly what you asked for, in that it doesn't do the reductions using only the defining Coxeter relations, but it uses longer substitutions, which can of course be derived from the defining relations, and are always length or lexicographically decreasing. It is the reduction algorithm based on the automatic structure of Coxeter groups, and runs in quadratic time in general. It has been implemented (for example) in the Computer Algebra System Magma.