Parametrization by polytopes of intersections of orbits by conjugation

Let S be an nXn real symmetric matrix with spectral decomposition S=Q^T Lambda Q, where Q is an orthogonal matrix and Lambda is diagonal with simple spectrum {lambda_1,..., lambda_n}. Also let O_S e R_S be the orbits by conjugation of S by, respectively, orthogonal matrices and upper triangular matrices with positive diagonal. Denote by F_S the intersection O_S and R_S. We show that the map F tha goes from the closure of F_S to R^n and takes S' = (Q')^T Lambda Q' to diag(Q' Lambda (Q')^T) is a smooth bijection onto its range P_S, the convex hull of some subset of the n! permuatations of (lambda_1, ..., lambda_n). We also find necessary and sufficient conditions for P_S to have n! vertices.