On the vanishing of the hyperdeterminant under certain symmetry conditions

Abstract

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 Wλ(X, K) 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, K) = ⊗ d−1 i=1 Wi (d−1,1)(V, K). We finally prove the vanishing of the hyperdeterminant of any F ∈ (⊗ λ≠(d),(d−1,1)) ⊕ Wi (d−1,1)(V, K). This improves the result in [10] and [1], where the same result was proved without this new last summand.