Bruhat decomposition

Bruhat decomposition is the name for the fact that $B\backslash G/B=W$, where $G$ is a reductive group, $B$ a Borel subgroup, and $W$ the Weyl group. Less canonically, one can write $G=BWB$.

In the case of the general linear group $G=G{L}_n$, $B$ is the group of nonsingular upper triangular matrices, and $W$ is the collection of permutation matrices (and is isomorphic to $S_n$). Any nonsingular matrix can thus be written uniquely as a product of an upper triangular matrix, a permutation matrix, and another upper triangular matrix.

Title	Bruhat decomposition
Canonical name	BruhatDecomposition
Date of creation	2013-03-22 15:43:15
Last modified on	2013-03-22 15:43:15
Owner	nerdy2 (62)
Last modified by	nerdy2 (62)
Numerical id	9
Author	nerdy2 (62)
Entry type	Theorem
Classification	msc 20-00