On the vanishing of the hyperdeterminant under certain symmetry conditions

Abstract

Given a vector space V over a field K whose characteris tic is coprime with d!, let us decompose the vector spa of multilinear forms V ∗ ⊗ (d) ... ⊗ V ∗ = λ Wλ(X, K) ac cording to the different partitions λ of d, i.e. the different representations of Sd. In this paper we first give a decom position W(d−1,1)(V, K) = d i=1 Wi (d−1,1)(V, K). We final 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.