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 $V^{⁎}\otimes \stackrel{\text{(}d)}{\dots }\otimes V^{⁎}={⨁}_{\lambda }{W}_{\lambda }(X,K)$ 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 $F\in \left({⨁}_{\lambda \ne \left(d\right),(d-1,1)}\right)\oplus W_{(d-1,1)}^i(V,K)$. This improves the result in [10] and [1], where the same result was proved without this new last summand.