What are the symmetric functions of the eigenvalues of various specializations of matrices?

Consider

• Hermitian matrices
• Positive-definite matrices
• The determinant of a skew-symmetric matrix is the square root of the pfaffian. Consider the symmetric functions of the eigenvalues of other skew-symmetric matrices.
• Look for a mapping between the case of determinant equals one and trace equals zero and see how this might relate the multiplicative bases (elementary, homogeneous, power) and the nonmultiplicative bases. And consider the combinatorial role of the exponential matrix function linking the Lie group and Lie algebra.