The hyperdeterminant of 3 x 3 x 2 arrays, and the simplest invariant of 4 x 4 x 2 arrays

TL;DR

This paper derives explicit formulas for the hyperdeterminant of 3x3x2 arrays and the simplest invariant of 4x4x2 arrays using Lie algebra representation theory and computational linear algebra.

Contribution

It provides the first explicit formulas for these invariants, detailing their polynomial structure, degree, number of monomials, and symmetry orbits.

Findings

Explicit formula for 3x3x2 hyperdeterminant with 16749 monomials

Explicit formula for 4x4x2 invariant with 14148 monomials

Use of Lie algebra representation theory and computational methods

Abstract

We use the representation theory of Lie algebras and computational linear algebra to obtain an explicit formula for the hyperdeterminant of a $3 \times 3 \times 2$ array: a homogeneous polynomial of degree 12 in 18 variables with 16749 monomials and 41 distinct integer coefficients; the monomials belong to 178 orbits under the action of $(S_3 \times S_3 \times S_2) \rtimes S_2$. We also obtain the simplest invariant for a $4 \times 4 \times 2$ array: a homogeneous polynomial of degree 8 in 32 variables with 14148 monomials and 13 distinct integer coefficients; the monomials belong to 28 orbits under $(S_4 \times S_4 \times S_2) \rtimes S_2$.