We know that $GL_n(\mathbb{C})$ can be decomposed as $GL_n(\mathbb{C}) = \bigsqcup_{w \in W}BwB$ where $B$ is the subgroup of upper triangular invertible matrices and $W$ is the Weyl group isomorphic to the symmetric group $S_n$ on $n$ letters with permutation matrices as representatives.
So my question is: How to describe $BwB$ accurately for each $w \in W$?
More percisely, if $x = (x_{ij}) \in BwB$ for some $w \in W$, there must be some relations among all the $x_{ij}$, how to find such relations?
Edit:So the conditions we need to describe the partition were already introduced in "Double Bruhat Cells and Total Positivity" by Fomin and Zelevinsky as Proposition 4.1. However, I still wonder if there's a direct way to caculate such conditions out by equations and transformations.