Let $G=\operatorname{GL}(n,\mathbb C)$. What follows can be put into a more general context, but I would like to first understand it for this case, the generalization is a second step.

For Zariski-almost all $x\in G$, there is a unique decomposition $x=lu$ with $l$ lower triangular and $u$ upper unipotent triangular. Now, I can precompose $x$ with some permutation (matrix) $\pi$ (an element of the Weyl group of $G$), i.e. permute the rows of $x$. For generic $x$, the matrix $\pi x$ will again have some LU-Decomposition $\pi x = l_\pi u_\pi$ with $l_\pi$ lower triangular and $u_\pi$ upper unipotent triangular.

We have an open immersion $\iota:L\times U\hookrightarrow G$ mapping $(l,u)\mapsto lu$, where $L$ is the Borel of lower triangular matrices and $U$ the unipotent radical of its opposite Borel, the variety of upper unipotent triangular matrices. Restricting to some open subset $V\subseteq L\times U$, we can consider the map $(l,u)\mapsto \pi.(l,u) := (l_\pi,u_\pi)$ for all $\pi\in\mathfrak S_n$ such that $\iota(\pi.(l,u))=l_\pi u_\pi= \pi lu$.

Since $\iota(\pi.\sigma.(l,u))= (l_\pi)_\sigma (u_\pi)_\sigma = \sigma l_\pi u_\pi = \sigma\pi lu = l_{\sigma\pi} u_{\sigma\pi} = \iota(\pi\sigma.(l,u))$ and because $\iota$ is an immersion, this defines a Group action on $V$, turning $\iota|_V$ into a $\mathfrak S_n$-equivariant map. Hence, the action of $\mathfrak S_n$ on $V$ is algebraic.

My question is: Is there any way to describe the open sets $L\cap V$ (resp. $U\cap V$) and the action $l\mapsto l_\pi$ (resp. $u\mapsto u_\pi$) of the permutation group $\mathfrak S_n$ on them? I have no intuition what-so-ever what they look like, and calculating examples was not very insightful.

Thanks a lot in advance for any pointers.

**Edit**: The $n=2$ case might still explain better what is going on. Observe that
$$\begin{pmatrix} a & b \\ c & d \end{pmatrix} =
\begin{pmatrix} 1 & 0 \\ ca^{-1} & 1 \end{pmatrix} \cdot
\begin{pmatrix} a & b \\ 0 & d-ba^{-1}c\end{pmatrix}
$$
Now, switching the rows maps
\begin{align*}
\begin{pmatrix} 1 & 0 \\ ca^{-1} & 1 \end{pmatrix} &\mapsto
\begin{pmatrix} 1 & 0 \\ ac^{-1} & 1 \end{pmatrix} \\
\begin{pmatrix} a & b \\ 0 & d-ba^{-1}c\end{pmatrix} &\mapsto
\begin{pmatrix} c & d \\ 0 & b-dc^{-1}a\end{pmatrix}
\end{align*}
so on the lower unipotent triangular matrices, in the nonvanishing locus of the lower left entry, we map
\begin{align*}
\begin{pmatrix} 1 & 0 \\ a & 1 \end{pmatrix} &\mapsto
\begin{pmatrix} 1 & 0 \\ a^{-1} & 1 \end{pmatrix}
\end{align*}
I don't see immediately what the action on the upper triangular matrices is.

knowhow $\mathfrak S_n$ acts on $L$ and $U$, this is what I am asking. $\endgroup$1more comment