The hyperdeterminant vanishes for all but two Schur Functors

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 $Γ^λV\otimes S_λV\subseteq V^{\otimes p}$ where $λ$ is a Young diagram with p boxes and $λ_2\geq 2$ or $λ_3\geq 1$.