# 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 Grassmannians: 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

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.)

## Examples

Let *G* be the general linear group **GL**_{n} of invertible
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
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 Grassmannians. 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 Borels one may intersect the Bruhat cells for each of them.

## 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