In mathematics, the Bruhat decomposition (introduced by François Bruhat for classical groups and by Claude Chevalley in general)
G
=
B
W
B
{\displaystyle G=BWB}
of certain algebraic groups
G
=
B
W
B
{\displaystyle G=BWB}
into cells can be regarded as a general expression of the principle of Gauss–Jordan elimination, which generically writes a matrix as a product of an upper triangular and lower triangular matrices—but with exceptional cases. It is related to the Schubert cell decomposition of flag varieties: see Weyl group for this.
More generally, any group with a (B, N) pair has a Bruhat decomposition.
Contents
Definitions
G
{\displaystyle G}
is a connected, reductive algebraic group over an algebraically closed field.
B
{\displaystyle B}
is a Borel subgroup of
G
{\displaystyle G}
W
{\displaystyle W}
is a Weyl group of
G
{\displaystyle G}
corresponding to a maximal torus of
B
{\displaystyle B}
.
The Bruhat decomposition of
G
{\displaystyle G}
is the decomposition
G
=
B
W
B
=
⨆
w
∈
W
Examples
Let
G
{\displaystyle G}
be the general linear group GLn of invertible
n
×
n
{\displaystyle n\times n}
matrices with entries in some algebraically closed field, which is a reductive group. Then the Weyl group
W
{\displaystyle W}
is isomorphic to the symmetric group
S
n
{\displaystyle S_{n}}
on
n
{\displaystyle n}
letters, with permutation matrices as representatives. In this case, we can take
B
{\displaystyle B}
to be the subgroup of upper triangular invertible matrices, so Bruhat decomposition says that one can write any invertible matrix
A
{\displaystyle A}
as a product
U
1
Geometry
The cells in the Bruhat decomposition correspond to the Schubert cell decomposition of flag varieties. The dimension of the cells corresponds to the length of the word
w
{\displaystyle w}
in the Weyl group. Poincaré duality constrains the topology of the cell decomposition, and thus the algebra of the Weyl group; for instance, the top dimensional cell is unique (it represents the fundamental class), and corresponds to the longest element of a Coxeter group.
Computations
The number of cells in a given dimension of the Bruhat decomposition are the coefficients of the
q
{\displaystyle q}
-polynomial of the associated Dynkin diagram.
Double Bruhat cells
With two opposite Borel subgroups, one may intersect the Bruhat cells for each of them, giving a further decomposition
G
=
⨆
w
1
,
w
2
∈
W
(
B
w
1
B
∩
B
−
w
2
B
−
)
.
{\displaystyle G=\bigsqcup _{w_{1},w_{2}\in W}(Bw_{1}B\cap B_{-}w_{2}B_{-}).}