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Gamas's theorem

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(Redirected from Gamas's Theorem) Mathematical 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 {\displaystyle 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 {\displaystyle V} be a finite-dimensional complex vector space and λ {\displaystyle \lambda } be a partition of n {\displaystyle n} . From the representation theory of the symmetric group S n {\displaystyle S_{n}} it is known that the partition λ {\displaystyle \lambda } corresponds to an irreducible representation of S n {\displaystyle S_{n}} . Let χ λ {\displaystyle \chi ^{\lambda }} be the character of this representation. The tensor v 1 ⊗ v 2 ⊗ ⋯ ⊗ v n ∈ V ⊗ n {\displaystyle v_{1}\otimes v_{2}\otimes \dots \otimes v_{n}\in V^{\otimes n}} symmetrized by χ λ {\displaystyle \chi ^{\lambda }} is defined to be

χ λ ( e ) n ! ∑ σ ∈ S n χ λ ( σ ) v σ ( 1 ) ⊗ v σ ( 2 ) ⊗ ⋯ ⊗ v σ ( n ) , {\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 e {\displaystyle e} is the identity element of S n {\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 { v i } {\displaystyle \{v_{i}\}} into linearly independent sets whose sizes are in bijection with the lengths of the columns of the partition λ {\displaystyle \lambda } .

See also

References

  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.
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