Bruhat decomposition theorem

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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