Bruhat decomposition
In mathematics, the Bruhat decomposition 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.
Definitions
- is a connected, reductive algebraic group over an algebraically closed field.
- is a Borel subgroup of
- is a Weyl group of corresponding to a maximal torus of.
of as a disjoint union of double cosets of parameterized by the elements of the Weyl group.
Examples
Let 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 on letters, with permutation matrices as representatives. In this case, we can take to be the subgroup of upper triangular invertible matrices, so Bruhat decomposition says that one can write any invertible matrix as a product where and are upper triangular, and is a permutation matrix. Writing this as, 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 to row if . The row operations correspond to, and the column operations correspond to.The special linear group SLn of invertible matrices with determinant is a semisimple group, and hence reductive. In this case, 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 instead of. Here 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.