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.^[1] Additional proofs have been given by Pate^[2] and Berget.^[3]

Statement of the theorem

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

$\frac{χ^{λ} (e)}{n!} \sum_{σ \in S_{n}} χ^{λ} (σ) v_{σ (1)} \otimes v_{σ (2)} \otimes \dots \otimes v_{σ (n)},$

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 $λ$ .

References

↑ Carlos Gamas (1988). "Conditions for a symmetrized decomposable tensor to be zero". Linear Algebra and Its Applications. 108. Elsevier: 83–119. doi:10.1016/0024-3795(88)90180-2.
↑ Thomas H. Pate (1990). "Immanants and decomposable tensors that symmetrize to zero". Linear and Multilinear Algebra. 28 (3). Taylor & Francis: 175–184. doi:10.1080/03081089008818039.
↑ Andrew Berget (2009). "A short proof of Gamas's theorem". Linear Algebra and Its Applications. 430 (2). Elsevier: 791–794. arXiv:0906.4769. doi:10.1016/j.laa.2008.09.027. S2CID 115172852.

[Gamas-1] Carlos Gamas (1988). "Conditions for a symmetrized decomposable tensor to be zero". Linear Algebra and Its Applications. 108. Elsevier: 83–119. doi:10.1016/0024-3795(88)90180-2.

[Pate-2] Thomas H. Pate (1990). "Immanants and decomposable tensors that symmetrize to zero". Linear and Multilinear Algebra. 28 (3). Taylor & Francis: 175–184. doi:10.1080/03081089008818039.

[Berget-3] Andrew Berget (2009). "A short proof of Gamas's theorem". Linear Algebra and Its Applications. 430 (2). Elsevier: 791–794. arXiv:0906.4769. doi:10.1016/j.laa.2008.09.027. S2CID 115172852.

[1]

[2]

[3]

Statement of the theorem

See also

References