Gamas's theorem

Gamas's theorem is a result in multilinear algebra which states the necessary and sufficient conditions for a tensor symmetrized by an irreducible representation of the symmetric group $$S_n$$ to be zero. It was proven in 1988 by Carlos Gamas. Additional proofs have been given by Pate and Berget.

Statement of the theorem
Let $$V$$ be a finite-dimensional complex vector space and $$\lambda$$ be a partition of $$n$$. From the representation theory of the symmetric group $$S_n$$ it is known that the partition $$\lambda$$ corresponds to an irreducible representation of $$S_n$$. Let $$\chi^{\lambda}$$ be the character of this representation. The tensor $$v_1 \otimes v_2 \otimes \dots \otimes v_n \in V^{\otimes n}$$ symmetrized by $$\chi^{\lambda}$$ is defined to be

$$ \frac{\chi^{\lambda}(e)}{n!} \sum_{\sigma \in S_n} \chi^{\lambda}(\sigma) v_{\sigma(1)} \otimes v_{\sigma(2)} \otimes \dots \otimes v_{\sigma(n)}, $$

where $$e$$ is the identity element of $$S_n$$. Gamas's theorem states that the above symmetrized tensor is non-zero if and only if it is possible to partition the set of vectors $$\{v_i\}$$ into linearly independent sets whose sizes are in bijection with the lengths of the columns of the partition $$\lambda$$.