# 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 |
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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 |