  The problem of computing the global minimum of a trigonometric polynomial is
computationally hard. We address this problem for the case, where the
polynomial is invariant under the exponential action of a finite group. The
strategy is to follow an established relaxation strategy in order to obtain a
converging hierarchy of lower bounds. Those bounds are obtained by numerically
solving semi-definite programs (SDPs) on the cone of positive semi-definite
Hermitian Toeplitz matrices, which is outlined in the book of Dumitrescu
[Dum07]. To exploit the invariance, we show that the group has an induced
action on the Toeplitz matrices and prove that the feasible region of the SDP
can be restricted to the invariant matrices, whilst retaining the same
solution. Then we construct a symmetry adapted basis tailored to this group
action, which allows us to block-diagonalize invariant matrices and thus reduce
the computational complexity to solve the SDP.
  The approach is in its generality novel for trigonometric optimization and
complements the one that was proposed as a poster at the ISSAC 2022 conference
[HMMR22] and later extended to [HMMR24]. In the previous work, we first used
the invariance of the trigonometric polynomial to obtain a classical polynomial
optimization problem on the orbit space and subsequently relaxed the problem to
an SDP. Now, we first make the relaxation and then exploit invariance.
  Partial results of this article have been presented as a poster at the ISSAC
2023 conference [Met23].
