Bruhat decomposition theorem
Contents
Statement
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.
Combinatorics
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.
Particular cases
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 |