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  "body": "\n## Example\n\nLet $(W, S)$ 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.\nFor $u, w \\in W$, write $u \\to w$ when $w = u t$ for some $t \\in T$ and $\\ell(u) \u003c \\ell(w)$.\nThe *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 $k \\ge 0$.\nIt is a partial order on $W$, graded by the length $\\ell$.\nBy 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 $u$.\n\nCheck in the [[symmetric group|0515A6D2Y1]] $S_3$ with $S = \\set{s_1, s_2}$, the adjacent transpositions.\nIts 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 $3$.\nEach element of length $2$ has a reduced word that contains both $s_1$ and $s_2$ as subwords.\nSo both $s_1$ and $s_2$ lie below both $s_1 s_2$ and $s_2 s_1$.\nSo 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.\n\n## Remarks\n\nThe Bruhat order contains the [[weak order|C8SEGT1AZS]]: in $S_3$ the weak order has only two of the four relations between the middle levels.\nFor $S_n$, 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.\n\n## Explanations\n\n### L1 human\n\nElements of a Coxeter group, such as permutations, can be written as words in simple generators.\nOne 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.\nIn $S_3$ this gives one element at the bottom, two on each middle level, and one at the top.\nEvery element of level one lies below every element of level two.\nThe order describes how Schubert cells sit in the closures of other Schubert cells, which is why it appears in geometry.\n",
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