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  "body": "\n## Statement\n\nLet $G$ be a connected reductive linear algebraic group over an algebraically closed field $k$.\nLet $B$ be a Borel subgroup of $G$, let $T \\subseteq B$ be a maximal torus, and let $W = N_G(T)/T$ be the Weyl group.\nFor $w \\in W$, the double coset $B \\dot w B$ does not depend on the choice of a representative $\\dot w \\in N_G(T)$ of $w$, and it is written $BwB$.\nThen $G$ is the disjoint union\n$$G = \\bigsqcup_{w \\in W} BwB.$$\n\n## Remarks\n\nFor the [[general linear group|0FCHE902D7]] $G = \\mathrm{GL}_n(k)$, take $B$ the invertible [[upper triangular matrices|0N4XY7Z3NN]], $T$ the invertible diagonal matrices, and $W$ the [[symmetric group|0515A6D2Y1]] $S_n$ realised by [[permutation matrices|08SP3DF8NB]].\nThen the theorem says that each invertible matrix is $b_1 w b_2$ with $b_1, b_2 \\in B$ and a unique permutation matrix $w$.\nThe proof in this case is Gaussian elimination: row and column operations by upper triangular matrices clear an invertible matrix down to a permutation matrix.\nThe permutation is unique because the ranks of the lower left corner submatrices are unchanged by these operations, and they determine it.\n\nExample: for $n = 2$, $W = \\set{1, s}$ with $s = \\begin{pmatrix} 0 \u0026 1 \\\\ 1 \u0026 0 \\end{pmatrix}$.\nA matrix $\\begin{pmatrix} a \u0026 b \\\\ c \u0026 d \\end{pmatrix}$ lies in $B$ when $c = 0$ and in $BsB$ when $c \\ne 0$, so $\\mathrm{GL}_2(k) = B \\sqcup BsB$.\n\nThe same decomposition holds for every group with a BN-pair, and this is the basis of the theory of [[buildings|6BQJ50NEJ0]] and of [[groups of Lie type|NPEBABQ49M]].\n\n## Explanations\n\n### L1 human\n\nThe Bruhat decomposition cuts a large matrix group into finitely many pieces.\nEach piece is labelled by one permutation, or more generally by one element of the Weyl group.\nFor invertible matrices, the label of a matrix is the permutation that Gaussian elimination with upper triangular operations leaves behind.\n\n### L2 human\n\nFor a finite field $\\F_q$, the decomposition counts the elements of $\\mathrm{GL}_n(\\F_q)$ cell by cell, and it gives the order of many finite [[groups of Lie type|NPEBABQ49M]].\nIn geometry, the images of the double cosets $BwB$ in the flag variety $G/B$ are the Schubert cells.\nThe closure order of the cells is the Bruhat order on $W$.\nAbstracting the two subgroups $B$ and $N_G(T)$ gives the notion of a BN-pair, whose geometric picture is a [[building|6BQJ50NEJ0]].\n",
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