Английская Википедия: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 <math>S_n</math> 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 <math>V</math> be a finite-dimensional complex vector space and <math>\lambda</math> be a partition of <math>n</math>. From the representation theory of the symmetric group <math>S_n</math> it is known that the partition <math>\lambda</math> corresponds to an irreducible representation of <math>S_n</math>. Let <math>\chi^{\lambda}</math> be the character of this representation. The tensor <math>v_1 \otimes v_2 \otimes \dots \otimes v_n \in V^{\otimes n}</math> symmetrized by <math>\chi^{\lambda}</math> is defined to be

<math display="block"> \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)}, </math>

where <math>e</math> is the identity element of <math>S_n</math>. 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 <math>\{v_i\}</math> into linearly independent sets whose sizes are in bijection with the lengths of the columns of the partition <math>\lambda</math>.

See also

• Algebraic combinatorics
• Immanant
• Schur polynomial