Complex

A complex is an edge-colored graph that generalizes the flag graphs of polytopes.

Definition
There exist several equivalent ways to define the idea of a complex.

Edge colored graph
A complex is an edge-colored graph with a vertex set $V$ and a sequence $A$ where $$A(i)$$ yields the edges with color $i$ such that:


 * For every color $i$ the every vertex is incident on exactly 1 edge of color $i$. This is equivalent to saying that the edge-coloring is proper.
 * If $$i \neq j$$ then the intersection $$A(i)\cup A(j)$$ is empty.
 * The graph $$(V,A)$$ is connected.

Permutation sequence
An $n$-complex is a set $$\Omega$$ and a sequence of permutations on $$\Omega$$, $$A : \{1,\dots,n\} \rightarrow \mathrm{Aut}(\Omega)$$ such that:


 * For every $$A(i)$$ in $A$, $$A(i)\circ A(i)=\mathrm{id}$$
 * For every $$A(i)$$ in $A$, there is no element of $$\Omega$$ which is a fixed point of $$A(i)$$.
 * If $$i \neq j$$ then for any element $$x\in\Omega$$, $$A(i)(x)\neq A(j)(x)$$.
 * The group generated by the permutations in $A$ is transitive on $$\Omega$$.

Elements


For a given $i$ the elements of a complex of rank $i$ correspond to the orbits of the group generated by permutations $$A(j)$$ where $$i\neq j$$.

Alternatively the elements of rank $i$ in a a complex correspond to the connected components in the graph with edges of color $i$ removed.

Incidence
Two elements are incident on each other if their connected components share a vertex.