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 $G=BWB$ 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 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 $U_{1}PU_{2}$ where $U_{1}$ and $U_{2}$ are upper triangular, and $P$ is a permutation matrix. Writing this as $P=U_{1}^{-1}AU_{2}^{-1}$ , 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 $U_{1}^{-1}$ , and the column operations correspond to $U_{2}^{-1}$ .

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

Notes

^ This Week's Finds in Mathematical Physics, Week 186

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

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