On the vanishing of the hyperdeterminant under certain symmetry conditions

TL;DR

This paper investigates the conditions under which the hyperdeterminant vanishes, focusing on specific symmetry conditions and decompositions of multilinear forms, extending previous results by including additional summands.

Contribution

It introduces a new decomposition of multilinear forms and proves the vanishing of the hyperdeterminant for a broader class of forms under symmetry conditions.

Findings

Hyperdeterminant vanishes for forms outside certain partitions.

Decomposition of multilinear forms into specific summands.

Extension of previous vanishing results to include new summands.

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 $V^*\otimes\overset{\text(d)}{\ldots}\otimes V^*=\bigoplus _\lambda W_\lambda(X,\K)$ according to the different partitions $\lambda$ of $d$, i.e. the different representations of $S_d$. In this paper we first give a decomposition $W_{(d-1,1)}(V,\K)=\bigoplus_{i=1}^{d-1}W_{(d-1,1)}^i(V,\K)$. We finally prove the vanishing of the hyperdeterminant of any $F\in(\bigoplus_{\lambda\ne(d),(d-1,1)})\oplus W_{(d-1,1)}^i(V,\K)$. This improves the result in [10] and [1], where the same result was proved without this new last summand.