Symmetry Adapted Gram Spectrahedra

This paper explores the geometric structure of spectrahedral cone, called symmetry adapted positive semidefinite (PSD) and Gram spectrahedron a symmetric polynomial. In particular, we determine dimension PSD describe its extremal rays, discuss matrix representations. We also consider spectrahedra for specific families polynomials including binary polynomials, quadratics, ternary quartics sextics which give us further insight into these sums squares polynomials.