Certifying the non-negativity of a symmetric polynomial is much easier than certifying the non-negativity of an arbitrary polynomial function: for instance, in (1) is proved that the complexity of deciding if a symmetric polynomial $f \in \mathbb{R}[X_1,\ldots,X_n]$ of fixed degree is non-negative grows only polynomially with $n$. This is in contrast to the general case which complexity is NP-hard already in the case of quartic forms (2).
Say that a polynomial is $G$-invariant if it is invariant under all actions of a linear group $G$. Are there any other results that reduces the complexity of deciding if a fixed degree $G$-invariant polynomial is non-negative for $G \ne S_{n}$?
(1): Riener, Cordian. "Symmetric semi-algebraic sets and non-negativity of symmetric polynomials." arXiv preprint [arXiv:1409.0699] (2014).
(2): Murty, Katta G., and Santosh N. Kabadi. "Some NP-complete problems in quadratic and nonlinear programming." Mathematical programming 39.2 (1987): 117-129.