Suppose you are given a linear combination of points of $\mathbb{Z}^n$ which corresponds to the weight datum of a nontrivial representation $V$ of some simple Lie algebra $g$. Is it possible to reconstruct $g$ and $V$ from this data, without knowing the canonical form on the weight lattice? (It looks like there might be some problems with dual Lie algebras  can you at least find the pair $g, g^{\vee}$?)

I'll write this as an answer rather than a comment to close the question. Victor Protsak's comment gives the original question a definitive answer of no. The formulation of the original question is indeed fuzzy, as is my knowledge of representation theory. I was interested in characters of (not necessarily simple) representations of simple Lie algebras as formal combinations of points in the full root lattice up to automorphisms of the lattice (ignoring the quadratic form). The question comes from number theory: we were trying to efficiently reconstruct the Zariski closure of the image of a Galois representation knowing its Frobenius polynomials at primes of good reduction. I'll close the question, but if the experts know a good answer, I'd be curious to know how far the indeterminacy can go (i.e., say, is there some relationship between all simple representations of a given simple algebra whose collections of weights are related by automorphisms of the root lattice). 

