Just one last question. It seems I cannot implement symmetry reduction properly while using the SumsofSquares package. Using your result, I can do symmetry reduction for SOS optimization problems without the SumsofSquares package. However, I would like to do symmetry reduction in SDSOS or DSOS optimization. I do not know how much you are familiar with these. SDSOS optimization is equivalent to scaled diagonally dominant programming which is a second-order cone programming rather than SDP, and DSOS is a linear programming.
Do you have any suggestions for how I can use your package for invariant SDSOS optimizations or invariant scaled diagonally dominant programming problems?