Gamas's Theorem

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

[math]\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)}, }[/math]

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

See also

• Algebraic combinatorics
• Immanant
• Schur polynomial