On the vanishing of the hyperdeterminant under certain symmetry conditions

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