Bruhat decomposition

In mathematics, the Bruhat decomposition (introduced by François Bruhat for classical groups and by Claude Chevalley in general) G = BWB of certain algebraic groups G 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

• G is a connected, reductive algebraic group over an algebraically closed field.
• B is a Borel subgroup of G
• W is a Weyl group of G corresponding to a maximal torus of B.

The Bruhat decomposition of G is the decomposition

$G=BWB=\bigsqcup _{w\in W}BwB$

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 because the maximal torus is contained in B.)

Examples

Let G be the general linear group GL_n of invertible $n\times n$ matrices with entries in some algebraically closed field, which is a reductive group. Then the Weyl group W is isomorphic to the symmetric group S_n on n letters, with permutation matrices as representatives. In this case, we can take B to be the subgroup of upper triangular invertible matrices, so Bruhat decomposition says that one can write any invertible matrix A as a product U₁PU₂ where U₁ and U₂ are upper triangular, and P is a permutation matrix. Writing this as P = U₁⁻¹AU₂⁻¹, 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 i (resp. column i) to row j (resp. column j) if i > j (resp. i < j). The row operations correspond to U₁⁻¹, and the column operations correspond to U₂⁻¹.

The special linear group SL_n of invertible $n\times n$ matrices with determinant 1 is a semisimple group, and hence reductive. In this case, W is still isomorphic to the symmetric group S_n. However, the determinant of a permutation matrix is the sign of the permutation, so to represent an odd permutation in SL_n, we can take one of the nonzero elements to be −1 instead of 1. Here B is the subgroup of upper triangular matrices with determinant 1, so the interpretation of Bruhat decomposition in this case is similar to the case of GL_n.

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 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.

Double Bruhat cells

With two opposite Borel subgroups, one may intersect the Bruhat cells for each of them, giving a further decomposition

$$G=\bigsqcup _{w_{1},w_{2}\in W}(Bw_{1}B\cap B_{-}w_{2}B_{-}).$$

See also

• Lie group decompositions
• Birkhoff factorization, a special case of the Bruhat decomposition for affine groups.
• Cluster algebra

References

• Borel, Armand. Linear Algebraic Groups (2nd ed.). New York: Springer-Verlag, 1991. ISBN 0-387-97370-2.
• Bourbaki, Nicolas, Lie Groups and Lie Algebras: Chapters 4–6 (Elements of Mathematics), Springer-Verlag, 2008. ISBN 3-540-42650-7