Let $S$ be a bounded geometric shape 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 $G(S)$ 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 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 are finite Coxeter groups. Some finite Coxeter groups are symmetry groups of just semiregular polytopes. Then, a positive answer to Question 2 would imply that every finite Coxeter group admits an irreducible faithful complex representation.