On the vanishing of the hyperdeterminant under certain symmetry conditions

Given a vector space V over a field K whose characteristic is coprime with d !, let us decompose the vector space of multilinear forms V ⁎ ⊗ … ( d ) ⊗ V ⁎ = ⨁ λ W λ ( X , K ) according to the different partitions λ of d , i.e. the different representations of S d . In this paper we first give a decomposition W ( d − 1 , 1 ) ( V , K ) = ⨁ i = 1 d − 1 W ( d − 1 , 1 ) i ( V , K ) . We finally prove the vanishing of the hyperdeterminant of any F ∈ ( ⨁ λ ≠ ( d ) , ( d − 1 , 1 ) ) ⊕ W ( d − 1 , 1 ) i ( V , K ) . This improves the result in [10] and [1] , where the same result was proved without this new last summand.