  This paper explores the geometric structure of the spectrahedral cone, called
the symmetry adapted PSD cone, and the symmetry adapted Gram spectrahedron of a
symmetric polynomial. In particular, we determine the dimension of the symmetry
adapted PSD cone, describe its extreme rays, and discuss the structure of its
matrix representations. We also consider the symmetry adapted Gram spectrahedra
for specific families of symmetric polynomials including binary symmetric
polynomials, quadratics, and ternary quartics and sextics which give us further
insight into these symmetric SOS polynomials. Finally, we discuss applications
of the theory of sums of squares and symmetric polynomials which arise from
symmetric function inequalities.
