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Let $P\subset\Bbb R^d$ be a vertex-transitive polytope aka. an orbit polytope.

Can there be a matrix $T\in\mathrm{SO}(\Bbb R^d)$ that commutes with all symmetries in $\mathrm{Aut}(P)\subset\mathrm O(\Bbb R^d)$?

Probably one approach to the question is as follows: can there be such a polytope $P\subset\Bbb R^d$ for which $\mathrm{Aut}(P)$ is (real) irreducible, but not absolutely irreducible (that is, not irreducible over $\Bbb C$). Vertex-transitivity is necessary for the question, as e.g. there is a polytope (not vertex-transitive) whose symmetry group is a finite subgroup of $\mathrm{SO}(\Bbb R^2)$, which is real irreducible, but reducible over $\Bbb C$.

Let $P\subset\Bbb R^d$ be a vertex-transitive polytope aka. an orbit polytope. Can there be a matrix $T\in\mathrm{SO}(\Bbb R^d)$ that commutes with all symmetries in $\mathrm{Aut}(P)\subset\mathrm O(\Bbb R^d)$?

Probably one approach to the question is as follows: can there be a vertex-transitive polytope $P\subset\Bbb R^d$ for which $\mathrm{Aut}(P)$ is (real) irreducible, but not absolutely irreducible (that is, not irreducible over $\Bbb C$).

Vertex-transitivity is necessary for all these questions. For example, there is a polytope (not vertex-transitive) whose symmetry group is a finite subgroup of $\mathrm{SO}(\Bbb R^2)$, which is real irreducible, but reducible over $\Bbb C$. Since $\mathrm{SO}(\Bbb R^2)$ is commutative, every element of that group would then commute with $\mathrm{Aut}(P)$. It is known that most commutative groups cannot be symmetry groups of vertex-transitive polytopes (only exceptions are elementary 2-abelian groups).

Let $P\subset\Bbb R^d$ be a vertex-transitive polytope aka. an orbit polytope.

Can there be a matrix $T\in\mathrm{SO}(\Bbb R^d)$ that commutes with all symmetries in $\mathrm{Aut}(P)\subset\mathrm O(\Bbb R^d)$?

Probably one approach to the question is as follows: can there be such a polytope $P\subset\Bbb R^d$ for which $\mathrm{Aut}(P)$ is (real) irreducible, but not absolutely irreducible (that is, not irreducible over $\Bbb C$).

Let $P\subset\Bbb R^d$ be a vertex-transitive polytope aka. an orbit polytope.

Can there be a matrix $T\in\mathrm{SO}(\Bbb R^d)$ that commutes with all symmetries in $\mathrm{Aut}(P)\subset\mathrm O(\Bbb R^d)$?

Probably one approach to the question is as follows: can there be such a polytope $P\subset\Bbb R^d$ for which $\mathrm{Aut}(P)$ is (real) irreducible, but not absolutely irreducible (that is, not irreducible over $\Bbb C$). Vertex-transitivity is necessary for the question, as e.g. there is a polytope (not vertex-transitive) whose symmetry group is a finite subgroup of $\mathrm{SO}(\Bbb R^2)$, which is real irreducible, but reducible over $\Bbb C$.

Let $P\subset\Bbb R^d$ be a vertex-transitive polytope aka. an orbit polytope.

Can there be a matrix $T\in\mathrm{SO}(\Bbb R^d)$ that commutes with all symmetries in $\mathrm{Aut}(P)\subset\mathrm O(\Bbb R^d)$?

Let $P\subset\Bbb R^d$ be a vertex-transitive polytope aka. an orbit polytope.

Can there be a matrix $T\in\mathrm{SO}(\Bbb R^d)$ that commutes with all symmetries in $\mathrm{Aut}(P)\subset\mathrm O(\Bbb R^d)$?

Probably one approach to the question is as follows: can there be such a polytope $P\subset\Bbb R^d$ for which $\mathrm{Aut}(P)$ is (real) irreducible, but not absolutely irreducible (that is, not irreducible over $\Bbb C$).