Let $S$ be a bounded [*geometric shape*][1] 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][2] $G(S)$ of $S$, and assume that
$$G(S)= \{g \in O(n) \ | \ g(S) = S  \}.$$  

Let $V:= E \otimes_{\mathbb{R}} \mathbb{C}$ be the [complexification][3] of $E$. It is a faithful complex representation of $G(S)$.  

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

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

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

Let's assume that $n>1$, $G(S) \neq \{ 1 \}$  and $ \langle S \rangle = E$.    


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

*Remark*: All symmetry group of [regular polytopes][4] are [finite Coxeter groups][5]. Some finite Coxeter groups are symmetry groups of just [semiregular polytopes][6]. Then, a positive answer to Question 2 would imply that every finite Coxeter group admits an irreducible faithful complex representation. 


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