Hyperdeterminants from the $E_8$ Discriminant

TL;DR

This paper derives explicit polynomial expressions for dual varieties of certain Grassmannians and hyperdeterminants, utilizing invariant theory, interpolation, and rational reconstruction, advancing algebraic geometry and invariant theory understanding.

Contribution

It provides new explicit formulas for dual varieties of Grassmannians and hyperdeterminants, connecting them through invariant polynomials and computational techniques.

Findings

Explicit polynomial formulas for dual Grassmannians

Expressions for hyperdeterminants of formats 3x3x3 and 2x2x2

Use of interpolation and rational reconstruction methods

Abstract

We find expressions of the polynomials defining the dual varieties of Grassmannians $Gr(3,9)$ and $Gr(4,8)$ both in terms of the fundamental invariants and in terms of a generic semi-simple element. We project the polynomial defining the dual of the adjoint orbit of $E_{8}$, and obtain the polynomials of interest as factors. To find an expression of the $Gr(4,8)$ discriminant in terms of fundamental invariants, which has $15,942$ terms, we perform interpolation with mod-$p$ reduction and rational reconstruction. From these expressions for the discriminants of $Gr(3,9)$ and $Gr(4,8)$ we also obtain expressions for well-known hyperdeterminants of formats $3\times 3\times 3$ and $2\times 2\times 2\times 2$.