Bruhat decomposition

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

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

The Bruhat decomposition of <math>G</math> is the decomposition

<math>G=BWB =\bigsqcup_{w\in W}BwB</math>

of <math>G</math> as a disjoint union of double cosets of <math>B</math> parameterized by the elements of the Weyl group <math>W</math>. (Note that although <math>W</math> is not in general a subgroup of <math>G</math>, the coset <math>wB</math> is still well defined because the maximal torus is contained in <math>B</math>.)

Examples

Let <math>G</math> be the general linear group GL_n of invertible <math>n \times n</math> matrices with entries in some algebraically closed field, which is a reductive group. Then the Weyl group <math>W</math> is isomorphic to the symmetric group <math>S_n</math> on <math>n</math> letters, with permutation matrices as representatives. In this case, we can take <math>B</math> to be the subgroup of upper triangular invertible matrices, so Bruhat decomposition says that one can write any invertible matrix <math>A</math> as a product <math>U_1PU_2</math> where <math>U_1</math> and <math>U_2</math> are upper triangular, and <math>P</math> is a permutation matrix. Writing this as <math>P=U_{1}^{-1}AU_{2}^{-1}</math>, 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 <math>i</math> (resp. column <math>i</math>) to row <math>j</math> (resp. column <math>j</math>) if <math>i>j</math> (resp. <math>i<j</math>). The row operations correspond to <math>U_{1}^{-1}</math>, and the column operations correspond to <math>U_{2}^{-1}</math>.

The special linear group SL_n of invertible <math>n \times n</math> matrices with determinant <math>1</math> is a semisimple group, and hence reductive. In this case, <math>W</math> is still isomorphic to the symmetric group <math>S_n</math>. 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 <math>-1</math> instead of <math>1</math>. Here <math>B</math> is the subgroup of upper triangular matrices with determinant <math>1</math>, 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 <math>w</math> 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 <math>q</math>-polynomial<ref>This Week's Finds in Mathematical Physics, Week 186</ref> 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 <math display="block">G=\bigsqcup_{w_1 , w_2\in W} ( Bw_1 B \cap B_- w_2 B_- ).</math>

See also

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

Notes

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References

• Borel, Armand. Linear Algebraic Groups (2nd ed.). New York: Springer-Verlag, 1991. Template:Isbn.
• Bourbaki, Nicolas, Lie Groups and Lie Algebras: Chapters 4–6 (Elements of Mathematics), Springer-Verlag, 2008. Template:Isbn