How do the Bruhat cells of based on upper triangular matrices relate to the cells of lower triangular matrices?

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

• 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) – mi.f.zh Dec 3 '20 at 4:20

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 $$GL_n$$).