A Short Proof of Gamas's Theorem

If \chi^\lambda is the irreducible character of the symmetric group S_n corresponding to the partition \lambda of n then we may symmetrize a tensor v_1 \otimes ... \otimes v_n by \chi^\lambda. Gamas's theorem states that the result is not zero if and only if we can partition the set {v_i} into linearly independent sets whose sizes are the parts of the transpose of \lambda. We give a short and self-contained proof of this fact.