Open cell decomposition after applying a Weyl group element

Question

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.

Answer 1

In lieu of any other answers till now: the question reminded me somewhat vaguely of "cell multiplication" in a Bruhat decomposition, namely, in a Bruhat decomposition $G=\bigsqcup_{w\in W} BwB$ of $G=GL_n(k)$ with $B$ a minimal parabolie (Borel), and $W$ (representatives for) the (spherical) Weyl group, the "cells" $BwB$ have a kind of multiplication rule. For $S\subset W$ the collection of reflections attached to simple roots (determined by choice of $B$), and for $\ell(w)$ the length-of-word function associated to generators $S$ for $W$, for $s\in S$, $w\in W$, $BsB\cdot BwB=BswB$ when $\ell(sw)>\ell(w)$, and $BsB\cdot BwB=BswB\cup BwB$ when $\ell(sw)<\ell(w)$.

One exposition of such stuff, minimizing reliance upon general Coxeter-group stuff, is at http://www.math.umn.edu/~garrett/m/v/bldgs.pdf

The standard treatment, going back many decades, and abstracted by Bruhat-Tits and others, does rely on the combinatorial group theory of Coxeter groups (and for not-split algebraic groups, on the more serious theory of algebraic groups).