In mathematics, the Bruhat decomposition (introduced by François Bruhat for classical groups and by Claude Chevalley in general)
of certain algebraic groups
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.
The Bruhat decomposition of
as a disjoint union of double cosets of
parameterized by the elements of the Weyl group
is still well defined because the maximal torus is contained in
be the general linear group GLn of invertible
matrices with entries in some algebraically closed field, which is a reductive group.
Then the Weyl group
is isomorphic to the symmetric group
letters, with permutation matrices as representatives.
to be the subgroup of upper triangular invertible matrices, so Bruhat decomposition says that one can write any invertible matrix
, this says that any invertible matrix can be transformed into a permutation matrix via a series of row and column operations, where we are only allowed to add row
The row operations correspond to
, and the column operations correspond to
The special linear group SLn of invertible
is a semisimple group, and hence reductive.
is still isomorphic to the symmetric group
However, the determinant of a permutation matrix is the sign of the permutation, so to represent an odd permutation in SLn, we can take one of the nonzero elements to be
is the subgroup of upper triangular matrices with determinant
, so the interpretation of Bruhat decomposition in this case is similar to the case of GLn.
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
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.
The number of cells in a given dimension of the Bruhat decomposition are the coefficients of the
With two opposite Borel subgroups, one may intersect the Bruhat cells for each of them, giving a further decomposition