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 $\lambda$.

See also

• Algebraic combinatorics
• Immanant
• Schur polynomial