Gamas's theorem explained

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 \chi ^{λ} be the character of this representation. The tensor v _{1} ⊗ v _{2} ⊗ ... ⊗ v _{n} \in V ^{⊗} symmetrized by \chi ^{λ} is defined to be

$$\backslash frac\; \backslash sum\_\; \backslash chi^(\backslash sigma)\; v\_\; \backslash otimes\; v\_\; \backslash otimes\; \backslash dots\; \backslash otimes\; v\_,$$

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 .

See also

• Algebraic combinatorics
• Immanant
• Schur polynomial

References

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