On the vanishing of the hyperdeterminant under certain symmetry conditions

Given a vector space $V$ over a field $\text{K}$ whose characteristic is coprime with $d!$, let us decompose the vector space of multilinear forms $V^{\ast }\otimes \otimes V^{\ast }={⨁}_{\lambda }{W}_{\lambda }(X,\text{K})$ according to the different partitions $\lambda$ of $d$, i.e. the different representations of $S_d$. In this paper we first give a decomposition $W_{(d-1,1)}(V,\text{K})={⨁}_{i=1}^{d-1}{W}_{(d-1,1)}^i(V,\text{K})$. We finally prove the vanishing of the hyperdeterminant of any $F\in ({⨁}_{\lambda \mathrm{\ne }(d),(d-1,1)})\oplus W_{(d-1,1)}^i(V,\text{K})$. This improves the result in [10] and [1], where the same result was proved without this new last summand.