Bruhat decomposition theorem
The Bruhat decomposition is a decomposition of a connected reductive linear algebraic group over a field . We denote by a Borel subgroup of and by the Weyl group corresponding to a maximal torus contained in that Borel subgroup in .
Note on algebraically closed: If is algebraically closed, then the Bruhat decomposition, up to isomorphism, depends only on the choice of and not on the specification of and . This essentially follows from the Borel-Morozov theorem. If is not algebraically closed, we need to explicitly specify the and that are being used.
The standard case of interest is where denotes the general linear group: the group of invertible matrices over . Let denote the Borel subgroup of : the subgroup comprising the upper triangular invertible matrices. Let denote the symmetric group of degree viewed as a group of permutation matrices, viewed as matrices over . Note that is a subgroup of when viewed this way.
Double coset formulation
The elements of the Weyl group form representatives for the double coset space of in .
Partition of space of complete flags
The double cosets of in can be identified with the sets of left cosets that live in them. Converting the double coset formulation into this language, we have:
where each can be thought of as the subset of the left coset space .
In the case with the corresponding choices of and , the set is identified with the set of complete flags of subspaces in -dimensional space over .
Formulation in terms of actions
Consider the left coset space . acts on this by left multiplication. Now, consider the induced diagonal action on . Further, since is a subgroup of , we get a corresponding action of on by left multiplication.
The claim is that the orbits under this action can be identified with the elements of . More explicitly, each orbit has a unique representative of the form:
Here, 1 stands for the identity element of .
Further, every element of the above form occurs in exactly one orbit.
The combinatorial version of the Bruhat decomposition theorem considers:
and identifies each of the pieces as a Schubert cell.
For the case that is a finite field of size and we are considering , the size of each cell is a polynomial in . The Weyl group is the symmetric group . The polynomial itself depends only on the corresponding element of and not on . In fact, it turns out that this polynomial is just where is the Bruhat word length. The sum of all these polynomials is the polynomial giving the size of , which is:
Plugging in in this expression gives , the order of the symmetric group.
General linear group
|Value of||Corresponding symmetric group||Corresponding general linear group||Discussion of Bruhat decomposition|
|1||trivial group||multiplicative group of||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|