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 $G(S) \le O(n)$.

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: The symmetry group of any regular polytope is an irreducible finite Coxeter group, and every such group is of this form, except those of type $D_n$, $E_6$, $E_7$, and $E_8$ which are symmetry groups of semiregular polytopes.