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 $\mathrm{Aut}(P)\subset\mathrm O(\Bbb R^d)$?
A matrix that commutes with all symmetries of a vertex-transitive polytope
M. Winter
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