Bruhat decomposition theorem

Statement

The Bruhat decomposition is a decomposition of a connected reductive linear algebraic group $G$ over a field $K$. We denote by $B$ a Borel subgroup of $G$ and by $W$ the Weyl group corresponding to a maximal torus contained in that Borel subgroup in $G$.

Note on algebraically closed: If $K$ is algebraically closed, then the Bruhat decomposition, up to isomorphism, depends only on the choice of $G$ and not on the specification of $B$ and $W$. This essentially follows from the Borel-Morozov theorem. If $K$ is not algebraically closed, we need to explicitly specify the $B$ and $W$ that are being used.

The standard case of interest is where $G = GL(n,K)$ denotes the general linear group: the group of invertible $n \times n$ matrices over $K$. Let $B = B(n,K)$ denote the Borel subgroup of $GL(n,K)$: the subgroup comprising the upper triangular invertible matrices. Let $W = S_n$ denote the symmetric group of degree $n$ viewed as a group of permutation matrices, viewed as matrices over $K$. Note that $S_n$ is a subgroup of $GL(n,K)$ when viewed this way.

Double coset formulation

The elements of the Weyl group form representatives for the double coset space of $B$ in $G$.

$G = \bigsqcup_{w \in W} BwB$.

Partition of space of complete flags

The double cosets of $B$ in $G$ can be identified with the sets of left cosets that live in them. Converting the double coset formulation into this language, we have:

$G/B = \bigsqcup_{w \in W} BwB/B$

where each $BwB/B$ can be thought of as the subset of the left coset space $G/B$.

In the case $G = GL(n,K)$ with the corresponding choices of $B$ and $W$, the set $G/B$ is identified with the set of complete flags of subspaces in $n$-dimensional space over $k$.

Formulation in terms of actions

Consider the left coset space $G/B$. $G$ acts on this by left multiplication. Now, consider the induced diagonal action on $G/B \times G/B$. Further, since $B$ is a subgroup of $G$, we get a corresponding action of $B$ on $G/B \times G/B$ by left multiplication.

The claim is that the orbits under this action can be identified with the elements of $W$. More explicitly, each orbit has a unique representative of the form:

$(1,w), w \in W$

Here, 1 stands for the identity element of $W$.

Further, every element of the above form occurs in exactly one orbit.

Combinatorics

The combinatorial version of the Bruhat decomposition theorem considers:

$G/B = \bigsqcup_{w \in W} BwB/B$

and identifies each of the pieces $BwB/B$ as a Schubert cell.

For the case that $K$ is a finite field of size $q$ and we are considering $G = GL(n,K)$, the size of each cell is a polynomial in $q$. The Weyl group $W$ is the symmetric group $S_n$. The polynomial itself depends only on the corresponding element of $S_n$ and not on $q$. In fact, it turns out that this polynomial is just $q^d$ where $d$ is the Bruhat word length. The sum of all these polynomials is the polynomial giving the size of $G/B$, which is:

$\prod_{i=1}^n \frac{q^i - 1}{q - 1} = \prod_{i=2}^n (\sum_{j=0}^{i-1} q^j)$

Plugging in $q = 1$ in this expression gives $n!$, the order of the symmetric group.

Particular cases

General linear group

Value of $n$	Corresponding symmetric group $S_n$	Corresponding general linear group $GL(n,K)$	Discussion of Bruhat decomposition
1	trivial group	multiplicative group of $K$	only one piece, not interesting
2	cyclic group:Z2	general linear group of degree two	Bruhat decomposition for general linear group of degree two over a field
3	symmetric group:S3	general linear group of degree three	Bruhat decomposition for general linear group of degree three over a field
4	symmetric group:S4	general linear group of degree four	Bruhat decomposition for general linear group of degree four over a field