A characterization of the permanent function by the Binet-Cauchy theorem (Q1103020)

Let K be a field of characteristic 0. Denote by \(M_ n(K)\) the set of \(n\times n\) matrices with entries in K. One may interpret \(n\in K\) if necessary by writing n as \(\sum^{n}_{k=1}1\), e.g. let \(E\in M_ n(K)\) be the matrix with each entry equal to 1/n. Denote by \(G_ n\) the set of all sequences \(\omega =(\omega_ 1,\omega_ 2,...,\omega_ n)\) of integers such that \(1\leq \omega_ 1\leq \omega_ 2\leq...\leq \omega_ n\leq n\). The authors show that if \(f: M_ n(K)\to K\) is such that \[ f(AB)=\sum_{\omega \in G_ n}(1/\mu (\omega))f(A[1,...,n| \omega])f(B[\omega | 1,...,n]) \] with f(E)\(\neq 0\), then either \(f(A)=n!/n^ n\) or else \(f(A)=\phi\) (per A) where \(\phi\) is an isomorphism of K. Here for any \(C\in M_ n(K)\) and \(\alpha,\beta \in G_ n\), C[\(\alpha| \beta]\) is that member of \(M_ n(K)\) whose rows are indexed by \(\alpha\) and whose columns are indexed by \(\beta\) ; \(\mu (\omega)=\mu_ 1!\mu_ 2!...\mu_ n!\), where \(\mu_ i\) is the number of times i appears in \(\omega\), \(i=1,...,n\).