---
tag: A17Q3X6285
kind: example
name: Bruhat order
msc: [20F55, 06A07]
instance_of: [1C7TKGTB2F]
uses: [G6FND7X69J, 0KBTP8Q2FS, 0515A6D2Y1, C8SEGT1AZS]
sources:
  - url: https://en.wikipedia.org/wiki/Bruhat_order
same_as:
  wikidata: Q4978697
  wikipedia: Bruhat order
---

## Example

Let $(W, S)$ be a [[Coxeter system|G6FND7X69J]], 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|0KBTP8Q2FS]] for $w$ has a subword that is a reduced word for $u$.

Check in the [[symmetric group|0515A6D2Y1]] $S_3$ 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.

## Remarks

The Bruhat order contains the [[weak order|C8SEGT1AZS]]: in $S_3$ the weak order has only two of the four relations between the middle levels.
For $S_n$, the reflections are the transpositions, so $u \to w$ when $w$ comes from $u$ by swapping two entries and the number of inversions goes up.

## Explanations

### L1 human

Elements of a Coxeter group, such as permutations, can be written as words in simple generators.
One element is below another in the Bruhat order when you can get it by deleting letters from a shortest word of the other, and still have a shortest word.
In $S_3$ this gives one element at the bottom, two on each middle level, and one at the top.
Every element of level one lies below every element of level two.
The order describes how Schubert cells sit in the closures of other Schubert cells, which is why it appears in geometry.