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 $$G = GL(n,K)$$ denote 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.