Given a topological group $G$ and a subset $S$ of $G$ that topologically generates it, what are the conditions under which an $n$-dimensional linear representation of $G$ over an algebraically closed field $k$ can be constructed once we have specified a degree $n$ characteristic polynomial in $k[x]$ for each element of $S$?

Under which further conditions if any is such a representation uniquely determined? Is there an algorithm for constructing it from the collection of characteristic polynomials?

Finally what if we don't require $k$ to be algebraically closed?