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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: 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.

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.

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.

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: 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.

All symmetry groups of regular polytopes are finite Coxeter groups.
Some finite Coxeter groups are symmetry groups of just semiregular polytopes.

The following are the two first sentences of the Wikipedia page "Symmetry group":

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 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?

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.