# 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 $G=GL_{n}$, $B$ 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 BruhatDecomposition 2013-03-22 15:43:15 2013-03-22 15:43:15 nerdy2 (62) nerdy2 (62) 9 nerdy2 (62) Theorem msc 20-00