---
tag: T91E86D0RH
kind: statement
role: theorem
name: Bruhat decomposition
msc: [20G15, "20E42"]
uses: [0FCHE902D7, 0N4XY7Z3NN, 08SP3DF8NB, 0515A6D2Y1]
sources:
  - url: https://en.wikipedia.org/wiki/Bruhat_decomposition
same_as:
  wikidata: Q4978699
  wikipedia: Bruhat decomposition
---

## Statement

Let $G$ be a connected reductive linear algebraic group over an algebraically closed field $k$.
Let $B$ 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.
For $w \in W$, 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 $BwB$.
Then $G$ is the disjoint union
$$G = \bigsqcup_{w \in W} BwB.$$

## Remarks

For 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]].
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|6BQJ50NEJ0]] and of [[groups of Lie type|NPEBABQ49M]].

## Explanations

### L1 human

The Bruhat decomposition cuts a large matrix group into finitely many pieces.
Each piece is labelled by one permutation, or more generally by one element of the Weyl group.
For invertible matrices, the label of a matrix is the permutation that Gaussian elimination with upper triangular operations leaves behind.

### L2 human

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|NPEBABQ49M]].
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|6BQJ50NEJ0]].