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.

单词	BruhatDecomposition
释义	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.
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