The answer for vectors uses the theory of multi-symmetric polynomials and in particular
power sums. You can learn about that in Emmanuel Briand's thesis mentioned in myprevious MO answer https://mathoverflow.net/questions/89337/generalizing-the-fundamental-theorem-of-symmetric-polynomials/89410#89410
Often people attibute the theory to Weyl (in his book on classical groups), however it is much older, with major contributions by Poisson, Schlaffli and Junker.