Reduced expression in a Coxeter group

A reduced expression for $w\in W$ is a product of the fewest possible simple generators representing $w$. The number of factors is its Coxeter length $\ell (w)$.

If $w=s_1\cdots s_n$ is a reduced expression and $s$ is simple with $\ell (ws)<\ell (w)$, thenfor some index $j$. The analogous statement holds for multiplication on the left.

Matsumoto's theorem says that any two reduced expressions for one element of a Coxeter group are connected by a finite sequence of braid moves.

Every word in simple Coxeter generators can be reduced by braid moves and cancellations $ss\mapsto 1$. Consequently, a word is reduced exactly when no sequence of braid moves can make a cancellation possible.