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Wikipedia / Bruhat decomposition

In mathematics, the Bruhat decomposition (introduced by François Bruhat for classical groups and by Claude Chevalley 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

The Bruhat decomposition is the decomposition

as a disjoint union of double cosets.)

Examples

be the general linear group GLn matrices with entries in some algebraically closed field, which is a reductive group is isomorphic to the symmetric group letters, with permutation matrices.

The special linear group SLn matrices with determinant is a semisimple group. However, the determinant of a permutation matrix is the sign of the permutation, so to represent an odd permutation in SLn, so the interpretation of Bruhat decomposition in this case is similar to the case of GLn.

Geometry

The cells in the Bruhat decomposition correspond to the Schubert cell decomposition of flag varieties. The dimension of the cells corresponds to the length 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

-polynomial of the associated Dynkin diagram.

Double Bruhat cells

See also

Notes

References

  • Borel, Armand. Linear Algebraic Groups (2nd ed.). New York: Springer-Verlag, 1991. .

  • Bourbaki, Nicolas, Lie Groups and Lie Algebras: Chapters 4–6 (Elements of Mathematics), Springer-Verlag, 2008.