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 matrix that commutes with all symmetries of a vertex-transitive polytope
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.




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


