:= x_{{i_r}}({t_r})\cdots x_{{i_1}}({t_1})$$ is a dense subset of ${N}^{{w}}$. More precisely, $\mathbf{x}_{i_r,\ldots,i_1}$ gives a birational isomorphism from $(\C^*)^r$ to ${N}^{{w}}$.

Thus, if $f\in\C[{N}^{{w}}]$ then $f(\mathbf{x}_{i_r,\ldots,i_1}(\mathbf{t}))$ is a rational function of $\mathbf{t}:=(t_r,\ldots,t_1)$ which completely determines $f$. Conversely, following Berenstein, Fomin and Zelevinsky [@BFZ0], one can give explicit formulas in terms of generalized minors for expressing each $t_i$ as a rational function on $N^w$ [@GLSCA]. These *Chamber Ansatz formulas* will be explained in §\[sectCA\] below.

\[Example3.6\] We continue Example \[example1\]. Thus $\g = \slchap_2$ and $w=s_2s_1s_2s_1$. By Example \[Example2\] (b), we have $$N_+ = \left\{\begin{pmatrix}
a & b \\
c & d 
\end{pmatrix}
\mid
a, b, c, d \in \C[z],\ ad-bc=1,\ 
 a(0)=d(0)=1,\ c(0)=0 
\right\}.$$ Writing, for $x=
\begin{pmatrix}
a & b \\
c & d 
\end{pmatrix}\in N_+$, $$a=1+\sum_{k\ge 1}a_kz^k,\quad
b=\sum_{k\ge 0}b_kz^k,\quad
c=\sum_{k\ge 1}c_kz^k,\quad
d=1+\sum_{k\ge 1}d_kz^k,$$ we get regular functions $a_k(x), b_k(x), c_k(x), d_k(x)$ on $N_+$, which by restriction give regular functions on $N^w$. We have $$x_1(t) =  \begin{pmatrix}
1 & t \\
0 & 1 
\end{pmatrix},\qquad
x_2(t) =  \begin{pmatrix}
1 & 0 \\
tz & 1 
\end{pmatrix},$$ and, putting $\mathbf{x}_{2,1,2,1}(\mathbf{t}) := x_2(t_4)x_1(t_3)x_2(t_2)x_1(t_1)$, we calculate $$\mathbf{x}_{2,1,2,1}(\mathbf{t}) = 
\begin{pmatrix}
1 + t_2t_3z& t_1+t_3+t_1t_2t_3z \\
(t_2+t_4)z+t_2t_3t_4z^2 & 1 +(t_1t_2+t_1t_4+t_3t_4)z+t_1t_2t_3t_4z^2 
\end{pmatrix}.$$ Hence $$\begin{array}{ll}
a_1(\mathbf{x}_{2,1,2,1}(\mathbf{t})) = t_2t_3,\\[2mm]
