Sum of squares (SOS) relaxations are often used to certify nonnegativity of polynomials and are equivalent to solving a semidefinite program (SDP). The feasible region of the SDP for a given polynomial is the Gram Spectrahedron. For symmetric polynomials, there are reductions to the problem size that can be done using tools from representation theory. This gives rise to a smaller, more manageable spectrahedron, the Symmetry Adapted Gram Spectrahedron. With this machinery, we disprove a 2011 conjecture about the complete homogeneous symmetric polynomials. Specifically, we find an SOS counterexample to the claim that H_lambda <= H_mu if and only if mu dominates lambda in partition dominance order.
