Gamas's Theorem

Let ${\displaystyle V}$ be a finite-dimensional complex vector space and ${\displaystyle \lambda }$ be a partition of ${\displaystyle n}$. From the representation theory of the symmetric group ${\displaystyle S_{n}}$ it is known that the partition ${\displaystyle \lambda }$ corresponds to an irreducible representation of ${\displaystyle S_{n}}$. Let ${\displaystyle \chi ^{\lambda }}$ be the character of this representation. The tensor ${\displaystyle v_{1}\otimes v_{2}\otimes \dots \otimes v_{n}\in V^{\otimes n}}$ symmetrized by ${\displaystyle \chi ^{\lambda }}$ is defined to be

$${\displaystyle {\frac {\chi ^{\lambda }(e)}{n!}}\sum _{\sigma \in S_{n}}\chi ^{\lambda }(\sigma )v_{\sigma (1)}\otimes v_{\sigma (2)}\otimes \dots \otimes v_{\sigma (n)},}$$

where ${\displaystyle e}$ is the identity element of ${\displaystyle S_{n}}$. 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 ${\displaystyle \{v_{i}\}}$ into linearly independent sets whose sizes are in bijection with the lengths of the columns of the partition ${\displaystyle \lambda }$.

• Algebraic combinatorics
• Immanant
• Schur polynomial