Bruhat order

Example

Let $(W, S)$ be a Coxeter system, let $\ell(w)$ be the length of $w \in W$ with respect to $S$, and let $T = \set{w s w^{-1} : w \in W, s \in S}$ be the set of reflections. For $u, w \in W$, write $u \to w$ when $w = u t$ for some $t \in T$ and $\ell(u) < \ell(w)$. The Bruhat order is the reflexive and transitive closure of $\to$: $u \le w$ when there is a sequence $u = u_0 \to u_1 \to \cdots \to u_k = w$ with $k \ge 0$. It is a partial order on $W$, graded by the length $\ell$. By the subword property, $u \le w$ if and only if some reduced word for $w$ has a subword that is a reduced word for $u$.

Check in the symmetric group $S_3$ with $S = \set{s_1, s_2}$, the adjacent transpositions. Its elements are $e$ of length $0$, $s_1, s_2$ of length $1$, $s_1 s_2, s_2 s_1$ of length $2$, and $s_1 s_2 s_1$ of length $3$. Each element of length $2$ has a reduced word that contains both $s_1$ and $s_2$ as subwords. So both $s_1$ and $s_2$ lie below both $s_1 s_2$ and $s_2 s_1$. So the Bruhat order on $S_3$ has the bottom $e$, the top $s_1 s_2 s_1$, and all four relations between the two middle levels.