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Let $F$ be a continuous function on the space of $N\times N$ hermitian matrices $\mathcal{H}_N$ taking values in $\mathbb{C}$ which satisfies $F(UMU^*)=F(M)$ for all $M\in \mathcal{H}_N$ and $U\in U(N)$, where the later stands for the unitary group of $\mathbb{C}^N$.

The first examples coming in mind are the Trace, and the Determinant. Do you know is there exists a way to express $F$ in terms of these two functions ? Or maybe under stronger conditions than continuity for $F$ ?

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Every symmetric function of the eigenvalues has this property. The elementary symmetric polynomials in the eigenvalues (the coefficients of the characteristic polynomial) freely generate the ring of symmetric polynomials, and these are dense in the space of functions you're interested in.

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    But you can write the elementary symmetric polynomial as a trace of an appropriate exterior power of the matrix (and perhaps also using a sum of determinants using Cauchy-Binet type of stuff)2011-11-25