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
It was proven in 1988 by Carlos Gamas.
[1] Additional proofs have been given by Pate[2] and Berget.
be a finite-dimensional complex vector space and
λ
From the representation theory of the symmetric group
λ
corresponds to an irreducible representation of
λ
be the character of this representation.
λ
λ
λ
is the identity element of
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
into linearly independent sets whose sizes are in bijection with the lengths of the columns of the partition
λ