The hyperdeterminant vanishes for all but two Schur Functors

TL;DR

This paper investigates the conditions under which the hyperdeterminant of a tensor vanishes, demonstrating it vanishes for all skew-symmetric tensors of rank p ≥ 3 and for certain Young diagram-based tensor spaces.

Contribution

It proves that the hyperdeterminant vanishes for all skew-symmetric tensors with p ≥ 3 and for specific tensor spaces associated with Young diagrams with particular shapes.

Findings

Hyperdeterminant vanishes for all skew-symmetric tensors with p ≥ 3.

Hyperdeterminant vanishes on tensor spaces associated with Young diagrams with λ₂ ≥ 2 or λ₃ ≥ 1.

Identifies specific tensor subspaces where the hyperdeterminant is zero.

Abstract

We recall the notion of hyperdeterminant of a multidimensional matrix (tensor). We prove that if we restrict the hyperdeterminant to a skew-symmetric tensor $\wedge^p V\subseteq V^{\otimes p}$ with $p \geq 3$ then it vanishes. The hyperdeterminant also vanishes when we restrict it to the space $\Gamma^\lambda V\otimes S_\lambda V\subseteq V^{\otimes p}$ where $\lambda$ is a Young diagram with p boxes and $\lambda_2\geq 2$ or $\lambda_3\geq 1$.