Reduced expression in a Coxeter group

A reduced expression in a Coxeter group has the least possible Coxeter length among expressions in the generators of its Coxeter system.

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 $\ell (w)$.

For a group generated by involutions, the deletion condition says that every nonreduced word can be shortened without changing its value by deleting two of its letters.

Arrange the lengths of all consecutive subwords of a word in a triangular grid. Under the folding condition, the boundary between length ascents and descents must contain a folding square. Its equal opposite vertices identify two letters that can be deleted. Induction proves the deletion condition for involutory generators.

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.