A short proof of Gamas’s theorem

If χλ is the irreducible character of Sn corresponding to the partition λ of n then we may symmetrize a tensor v1⊗⋯⊗vn by χλ. Gamas’s theorem states that the result is not zero if and only if we can partition the set {vi} into linearly independent sets whose sizes are the parts of the transpose of λ. We give a short and self-contained proof of this fact.