  The arithmetic mean/geometric mean-inequality (AM/GM-inequality) facilitates
classes of non-negativity certificates and of relaxation techniques for
polynomials and, more generally, for exponential sums. Here, we present a first
systematic study of the AM/GM-based techniques in the presence of symmetries
under the linear action of a finite group. We prove a symmetry-adapted
representation theorem and develop techniques to reduce the size of the
resulting relative entropy programs. We study in more detail the complexity
gain in the case of the symmetric group. In this setup, we can show in
particular certain stabilization results. We exhibit several sequences of
examples in growing dimensions where the size of the problem stabilizes.
Finally, we provide some numerical results, emphasizing the computational
speed-up.
