Bruhat decomposition

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In mathematics, the Bruhat decomposition 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.

More generally, any group with a (B,N) pair has a Bruhat decomposition.

1 Definitions
2 Computations
3 References
4 See also
5 References

[edit] Definitions

G is a semisimple algebraic group over an algebraically closed field.
B is a Borel subgroup of G
W is a the Weyl group of G corresponding to a maximal torus of B.

The Bruhat decomposition of G is the decomposition

$G=BWB =\cup_{w\in W}BwB$

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.)