Bruhat decomposition

Remarks

For the general linear group $G = \mathrm{GL}_n(k)$, take $B$ the invertible upper triangular matrices, $T$ the invertible diagonal matrices, and $W$ the symmetric group $S_n$ realised by permutation matrices. Then 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 $w$. The proof in this case is Gaussian elimination: row and column operations by upper triangular matrices clear an invertible matrix down to a permutation matrix. The permutation is unique because the ranks of the lower left corner submatrices are unchanged by these operations, and they determine it.

Example: for $n = 2$, $W = \set{1, s}$ with $s = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$. A matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ lies in $B$ when $c = 0$ and in $BsB$ when $c \ne 0$, so $\mathrm{GL}_2(k) = B \sqcup BsB$.

The same decomposition holds for every group with a BN-pair, and this is the basis of the theory of buildings and of groups of Lie type.

Going deeper

For a student

For 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. In geometry, the images of the double cosets $BwB$ in the flag variety $G/B$ are the Schubert cells. The closure order of the cells is the Bruhat order on $W$. Abstracting the two subgroups $B$ and $N_G(T)$ gives the notion of a BN-pair, whose geometric picture is a building.