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 ${\displaystyle S_{n}}${\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]

Let ${\displaystyle V}${\displaystyle V} be a finite-dimensional complex vector space and ${\displaystyle \lambda }${\displaystyle \lambda } be a partition of ${\displaystyle n}${\displaystyle n}. From the representation theory of the symmetric group ${\displaystyle S_{n}}${\displaystyle S_{n}} it is known that the partition ${\displaystyle \lambda }${\displaystyle \lambda } corresponds to an irreducible representation of ${\displaystyle S_{n}}${\displaystyle S_{n}}. Let ${\displaystyle \chi ^{\lambda }}${\displaystyle \chi ^{\lambda }} be the character of this representation. The tensor ${\displaystyle v_{1}\otimes v_{2}\otimes \dots \otimes v_{n}\in V^{\otimes n}}${\displaystyle v_{1}\otimes v_{2}\otimes \dots \otimes v_{n}\in V^{\otimes n}} symmetrized by ${\displaystyle \chi ^{\lambda }}${\displaystyle \chi ^{\lambda }} is defined to be

$${\displaystyle {\frac {\chi ^{\lambda }(e)}{n!}}\sum _{\sigma \in S_{n}}\chi ^{\lambda }(\sigma )v_{\sigma (1)}\otimes v_{\sigma (2)}\otimes \dots \otimes v_{\sigma (n)},}$${\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 ${\displaystyle e}${\displaystyle e} is the identity element of ${\displaystyle S_{n}}${\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 ${\displaystyle \{v_{i}\}}${\displaystyle \{v_{i}\}} into linearly independent sets whose sizes are in bijection with the lengths of the columns of the partition ${\displaystyle \lambda }${\displaystyle \lambda }.

• Algebraic combinatorics
• Immanant
• Schur polynomial

• Algebraic combinatorics
• Theorems
• Multilinear algebra

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