Gamas's theorem

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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 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

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Let   be a finite-dimensional complex vector space and   be a partition of  . From the representation theory of the symmetric group   it is known that the partition   corresponds to an irreducible representation of  . Let   be the character of this representation. The tensor   symmetrized by   is defined to be

 

where   is the identity element of  . 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   into linearly independent sets whose sizes are in bijection with the lengths of the columns of the partition  .

See also

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References

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  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.
  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.
  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.