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