A matrix that commutes with all symmetries of a vertex-transitive polytope 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).
 A: There are examples of vertex-transitive polytopes with irreducible symmetry groups for which there are still a non-scalar transformations $T\in\mathrm O(\Bbb R^d)$ that commutes with all the symmetries of the polytopes.
Fix a group $G$ with the following properties:

*

*$G$ is neither abelian nor generalized dicyclic.

*the centralizer of every irreducible representation of $G$ is isomorphic to either $\Bbb C$ or $\Bbb H$.

Examples are contained among the finite subgroups of $\mathrm U(n)$ (for $n\ge 2$) or $\mathrm{SU}(2)$. The first property ensures that $G$ is not among the groups that are excluded as symmetry groups of vertex-transitive polytopes as determined in [1]. So there is a vertex-transitive polytope $P$ whose automorphism group can be considered as a matrix group representation of $G$. By the second assumption, the centralizer of this matrix group containst then more than just scalar matrices.

[1] E. Friese, F. Ladisch. "Classification of Affine Symmetry Groups of Orbit Polytopes"
