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Let $B_+$ be the subgroup of $GL(n)$ of upper triangular matrices and let $B_-$ be the subgroup of lower triangular matrices. Let $\pi \in S_n$ and $P_{\pi}$ be the corresponding permutation matrix.

Is there any relation between the sets $B_+P_{\pi}B_+$ and $B_-P_{\pi}B_-$? Are they equal to eachother? Disjoint? Thanks in advance

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    $\begingroup$ It helps to think about the Bruhat decomposition for $GL_n$ as a generalisation of Gaussian elimination/row reduction (I answered this here but you can also find it elsewhere on MO I think) $\endgroup$ – mi.f.zh Dec 3 '20 at 4:20
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Since any Borel ($B_+$ or $B_-$) contains the maximal torus, the sets you've written down are not disjoint. But by my comment, they do not coincide either (although all Borels are conjugate to each other - this also holds for more general groups beyond $GL_n$).

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