My instructor defined “simplicial action” as a (finite) group action on a simplicial complex (a triangulation of a cell complex) that satisfies:
$1)$ Action takes vertices to vertices, the action on the other simplicies is linear.
$2)$ If $g\sigma =\sigma$ for $g\in G$, $\sigma \in |K|$, then $g |_{|\sigma |}$ is the identity map.
$3)$ For any $g\in G$ and $\sigma \in |K|$, the action of $g$ is uniquely determined by its action on the vertex set.
My question is that what is linearity of the action? How can the action on the vertex set determines completely the whole action. More important than these what is this concept in mathematic terminology? How can I find and study this concept? I couldn’t find such definition on several algebraic topology books.
Perhaps this might help.
If you have a simplex $\sigma$ with vertex set $v_0,...,v_m$ and another simplex $\tau$ with vertex set $w_0,...,w_n$ then any function $f : \{v_0,...,v_m\} \to \{w_0,...,w_n\}$ has a unique extension to a simplicial map $F : \sigma \to \tau$, namely: $$F(t_0 v_0 + ... + t_m v_m) = t_0 f(v_0) + ... + t_m f(v_m) $$ It follows that if you have two simplicial complexes $X,Y$, and if you have a function from the vertex set of $X$ to the vertex set of $Y$, and if for every simplex $\sigma$ of $X$ there exists a simplex $\tau$ of $Y$ such that the funcation takes the vertex set of $\sigma$ to the vertex set of $\tau$, than that function extends uniquely to a simplicial map from $X$ to $Y$.