characters of the $KR$-modules. These graphs are not unique, and their extensions are of interest also (see section \[extensions\]).

Clearly, the cluster graphs ${\mathcal G}_{{\mathfrak{g}}}$ are difficult to draw as the rank becomes high. Therefore, in treating the higher-rank algebras, it is useful to adopt a pictorial notation which is similar in spirit to the bipartite graph notation of [@FZfour].

First, we define the notion of a compound mutation to be a product of $n$ distinct commuting mutations. Let $I=\{i_1,...,i_n\}$ be the index set of these mutations, and define $\mu^{(I)}=\mu_{i_1}\circ \cdots
\circ \mu_{i_n}$. The order of mutations is irrelevant by definition. We may also use the notation $\mu^{(I)}=\prod_{i\in I}
\mu_i$. Obviously, since $\mu_i^2=1$, we preserve the property that $(\mu^{(I)})^2=1$ for the product of commuting notations.

The edge:

(0,0)(20,7) (3,3)(17,3) (3,3)(14,0)[2]{}[(0,0)[.7]{}]{} (10,5)[$I$]{} (3,1)[$t$]{} (17,1)[$t'$]{}

corresponds to the compound mutation $t' = \mu^{(I)}(t)$. In the graph ${\mathcal G}_{{\mathfrak{g}}}$, this corresponds to an $n$-simplex with node $t$ at one end and node $t'$ at the other. It is a diamond in the case for ${\mathcal G}_{{{\mathfrak{sl}}}_3}$ or a cube in the case of ${\mathcal G}_{{{\mathfrak{sl}}}_4}$.

Thus the graph ${\mathcal G}_{{\mathfrak{g}}}$ for ${{\mathfrak{g}}}$ simply-laced with the sequence of nodes and compound mutations as follows:

