Bruhat decomposition

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.

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:$G=BWB\; =cup\_\{win\; W\}BwB$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.)

Computations

The number of cells in a given dimension of the Bruhat decomposition are the coefficients of the "q"-polynomial [ [http://math.ucr.edu/home/baez/week186.html This Week's Finds in Mathematical Physics, Week 186] ] of the associated Dynkin diagram.

References

ee also

* Lie group decompositions

References

*Bourbaki, Nicolas, "Lie Groups and Lie Algebras: Chapters 4-6 (Elements of Mathematics)", ISBN 3-540-42650-7