Abstract
We report on the geometric structure of symmetry adapted PSD cones and symmetry adapted Gram spectrahedra of symmetric polynomials. We determine the dimension of symmetry adapted PSD cones, describe its extreme rays, and discuss the structure of its matrix representations. We also focus on symmetry adapted Gram spectrahedra of symmetric binary forms, quadrics, ternary quartics and sextics. In particular, we characterize extreme points of these spectrahedra for symmetric binary forms that are of rank two, and we report what we know about the facial structure of the same spectrahedra. The talk will be based on two collaborations, one with Alex Heaton and Isabelle Shankar, and another one with Matthew Heid.