The structure of Gram matrices of sum of squares polynomials with restricted harmonic support

Some sum of squares (SOS) polynomials admit decomposition certificates, or positive semidefinite Gram matrices, with additional structure. In this work, we use the structure of Gram matrices to relate the representation theory of $SL(2)$ to $SO(2)$. Informally, we prove that if p is a sum of squares and lives in some of the invariant subspaces of $SO(2)$, then it has a positive semidefinite Gram matrix that lives in certain invariant subspaces of $SL(2)$. The tools used in the proof construction are of independent interest.