A hyperbolicity cone is said to be rank-one generated (ROG) if all its extreme rays have rank
one, where the rank is computed with respect the underlying hyperbolic polynomial. This is a
natural class of hyperbolicity cones which are strictly more general than the ROG spectrahedral
cones. In this work, we present a study of the automorphisms of ROG hyperbolicity cones and their
derivative relaxations. One of our main results states that the automorphisms of the derivative
relaxations are exactly the automorphisms of the original cone fixing a certain direction. As
an application, we completely determine the automorphisms of the derivative relaxations of the
nonnegative orthant and of the positive semidefinite matrices. More generally, we also prove
relations between the automorphisms of a spectral cone and the underlying permutation-invariant
set, which might be of independent interest.
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