S} F^k]$, where $C$ is a chamber of $\mathbf S$. A $k$-cell $[C<_{\mathbf S} F^k]$ is in the boundary of a $j$-cell $[D<_{\mathbf S} G^j]$, with $k<j$, if

-   $F^k<_{\mathbf S} G^j$,

-   the chambers $C$ and $D$ are contained in the same chamber of the sub-arrangement $$\{H_{{\mathbb{R}}}\in{{\mathcal{A}}}_{{\mathbb{R}}}\mid F\subset H_{{\mathbb{R}}}\}.$$

The previous conditions are equivalent to saying that $C$ is the chamber of ${{\mathcal{A}}}_{{\mathbb{R}}}$ “closest” to $D$ among those containing $F^k$ in their closure.

It is possible to realize $\cal{S}({{\mathcal{A}}}_{{\mathbb{R}}})$ inside ${\mathbb{C}}^n$ with explicitly given attaching maps of the cells (see [@Sal87]).

Graphs and tournaments {#sec:graphs_tournaments}
----------------------

We will give here a short summary of graph theory to fix notation. For a complete discussion we refer the reader to Chartrand and Lesniak [@Chartrand-Lesniak] and Moon [@Moon].

#### Graphs

We will only take oriented simple graphs into account. Hence, throughout the paper, a [*graph*]{} will be a pair $({{\cal V}},{{\cal E}})$, where ${{\cal V}}$ is the set of [*nodes*]{} and ${{\cal E}}$ is the set of [*arcs*]{}, such that each pair of nodes $\{p,q\}$ is connected by at most one arc (either ${\overrightarrow{pq}}$ or ${\overleftarrow{pq}}$). If the arc ${\overrightarrow{pq}}$ (or ${\overleftarrow{qp}}$) is in ${{\cal E}}$, the node $p$ is said to 