In mathematics, the Bruhat decomposition (introduced by François Bruhat for classical groups and by Claude Chevalley in general) 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 Grassmannians: see Weyl group for this.
Contents
More generally, any group with a (B,N) pair has a Bruhat decomposition.
Definitions
The Bruhat decomposition of G is the decomposition
of G as a disjoint union of double cosets of B parameterized by the elements of the Weyl group W. (Note that although W is not in general a subgroup of G, the coset wB is still well defined.)
Examples
Let G be the general linear group GLn of invertible
The special linear group SLn of invertible
Geometry
The cells in the Bruhat decomposition correspond to the Schubert cell decomposition of Grassmannians. The dimension of the cells corresponds to the length of the word 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-polynomial of the associated Dynkin diagram.