All [symmetry groups][1] of [regular polytopes][2] are [finite Coxeter groups][3].  
Some finite Coxeter groups are symmetry groups of just [semiregular polytopes][4].    

The following are the two first sentences of the Wikipedia page "[Symmetry group][1]": 
> In group theory, the symmetry group of an object (image, signal, etc.)
> is the group of all transformations under which the object is
> invariant with composition as the group operation. For a space with a
> metric, it is a subgroup of the isometry group of the space concerned.  

According to the following extract, the transformations can be chosen linear if the figure is bounded:

> Any symmetry group whose elements have a common fixed point, which is
> true for all finite symmetry groups and also for the symmetry groups
> of bounded figures, can be represented as a subgroup of the orthogonal
> group $O(n)$ by choosing the origin to be a fixed point.

Let $F$ be a bounded figure in the Euclidean space $E=\mathbb{R^n}$. Assume that the origin of $E$ is a fixed point of every element of the symmetry group $G(F)$ of $F$. Then:  
$$G(F)= \{g \in O(n) \ | \ g(F) = F  \}.$$  

Let $V:= E \otimes_{\mathbb{R}} \mathbb{C}$ be the [complexification][5] of $E$. It is a faithful complex representation of $G(F)$.  

**Question**: Under which conditions on $F$, the representation $V$ is irreducible?

*Remark*: Here are two cases where $V$ is not irreducible:   
   
- $G(F) = \{ 1 \}$ and $n>1$,  
- the vector space generated by $F$, denoted $ \langle F \rangle$,  is a strict subspace of $E$.

So, let's assume from now that $n>1$, $G(F) \neq \{ 1 \}$  and $ \langle F \rangle = E$.    


 *For people thinking my question too broad, here are more specific questions:*  

*Question 1*: Is $V$ irreducible if $F$ is a regular polytope?  
*Question 2*: If so, can we extend to semiregular polytope?  
*Question 3*: If so, what is your better extension?


  [1]: https://en.wikipedia.org/wiki/Symmetry_group
  [2]: https://en.wikipedia.org/wiki/Regular_polytope
  [3]: https://en.wikipedia.org/wiki/Coxeter_group#Finite_Coxeter_groups
  [4]: https://en.wikipedia.org/wiki/Semiregular_polytope
  [5]: https://en.wikipedia.org/wiki/Complexification