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 GLn 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 SLn 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 SLn, 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 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 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 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>