Gamas's Theorem

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 {displaystyle 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 {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 } .

See also Algebraic combinatorics Immanant Schur polynomial References ^ Carlos Gamas (1988). "Conditions for a symmetrized decomposable tensor to be zero". Linear Algebra and its Applications. Elsevier. 108: 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. Taylor & Francis. 28 (3): 175–184. doi:10.1080/03081089008818039. ^ Andrew Berget (2009). "A short proof of Gamas's theorem". Linear Algebra and its Applications. Elsevier. 430 (2): 791–794. doi:10.1016/j.laa.2008.09.027. Categories: Algebraic combinatoricsTheoremsMultilinear algebra

Si quieres conocer otros artículos parecidos a Gamas's Theorem puedes visitar la categoría Algebraic combinatorics.

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