  Let $C$ be an irreducible smooth complex projective curve, and let $E$ be an
algebraic vector bundle of rank $r$ on $C$. Associated to $E$, there are vector
bundles ${\mathcal F}_n(E)$ of rank $nr$ on $S^n(C)$, where $S^n(C)$ is $
$n$-th symmetric power of $C$. We prove the following: Let $E_1$ and $E_2$ be
two semistable vector bundles on $C$, with ${\rm genus}(C)\, \geq\, 2$. If
${\mathcal F}_n(E_1)\,= \, {\mathcal F}_n(E_2)$ for a fixed $n$, then $E_1
\,=\, E_2$.
