This is a question about Postnikov's theory of positroids and plabic graphs. The short version is
If we have an non-reduced plabic graph $G$, how do we look at the alternating strands and read off the decorated permutation of the positroid cell that $G$ redundantly parameterizes?
Here are the necessary definitions, to make sure we agree on vocabulary: Suppose that I have a bipartite planar graph $G$ embedded in a disk, with $n$ vertices on the boundary of the disk. Let the boundary vertices all be black; in the interior, let there be $m+k$ white vertices and $m$ black vertices. A perfect matching $M$ of $G$ is a configuration of edges which covers every interior vertex exactly once, and covers $k$ of the $n$ boundary vertices. The set of boundary vertices which are covered is called $\partial(M)$. We will assume that $G$ has at least one perfect matching.
Let a positive real weight $w(e)$ be assigned to each edge. For a matching $M$, let $w(M) = \prod_{e \in M} w(e)$. For a $k$-element subset $I$ of $[n]$, let $\Delta_I(G,w) = \sum_{\partial(M) = I} w(M)$. The $k$-tuple of numbers $\Delta_I(G,w)$, as $I$ varies, obey the Plucker relations, and thus give a point $\mu(G,w)$ of the Grassmannian $G(k,n)$. Moreover, it is a point where all the Plucker coordinates are nonnegative, which makes it a point of the totally nonnegative Grassmannian $G(k,n)_{\geq 0}$.
Postnikov divides $G(k,n)_{\geq 0}$ up into cells, indexed by decorated permutations. If we fix $G$ and let $w$ vary over $\mathbb{R}_{>0}^{\mathrm{Edge}(G)}$, then $\mu(G,w)$ varies over one such cell. I want to know how to read off each one. More precisely, I know that I can read this off from the data of which sets $I$ have matchings $M$ with $\partial(M) = I$. What I want is a description that looks something like Postnikov's "rules of the road", which works when $G$ is reduced.
An analogy: Consider the set of upper triangular matrices with $1$'s on the diagonal and all minors nonnegative. This is stratified into $n!$ cells, indexed by $S_n$. For $1 \leq i \leq n-1$, let $x_i(t)$ be the Chevalley matrix $\mathrm{Id}_n + t e_{i(i+1)}$, where $e_{ij}$ is $1$ in position $(i,j)$ and $0$ everywhere else. For any sequence of indices $i_1$, $i_2$, ..., $i_N$ and any $t_1$, $t_2$, ..., $t_N > 0$, the product $x_{i_1}(t_1) \cdots x_{i_N}(t_N)$ is totally nonnegative. If we fix the word $(i_1, \ldots, i_N)$ but let the $t$'s vary over $\mathbb{R}_{>0}^N$, the image is one of the $n!$ cells.
If $s_{i_1} \cdots s_{i_N}$ is a reduced word in $S_n$, then the image of $(t_1, t_2, \ldots, t_N) \mapsto x_{i_1}(t_1) \cdots x_{i_N}(t_N)$ is the cell indexed by the product $s_{i_1} \cdots s_{i_N}$. If not, this is not true: We need to use the "Demazure" or "$0$-Hecke" product on $S_n$ instead of the regular product. I want to know the analogue of the $0$-Hecke rule for plabic graphs.
