A hyperdeterminant for 2 x 2 x 3 arrays

TL;DR

This paper constructs a degree 6 invariant polynomial for 2 x 2 x 3 arrays using Lie algebra representation theory, generalizing Cayley's hyperdeterminant to larger arrays.

Contribution

It introduces the first nonconstant invariant polynomial for 2 x 2 x 3 arrays, extending the concept of hyperdeterminants beyond 2 x 2 x 2 arrays.

Findings

Invariant polynomial has degree 6 with 66 terms

Coefficients are 1, -1, 2, -2

Generalizes Cayley's hyperdeterminant

Abstract

We use the representation theory of Lie algebras and computational linear algebra to determine the simplest nonconstant invariant polynomial in the entries of a general 2 x 2 x 3 array. This polynomial is homogeneous of degree 6 and has 66 terms with coefficients 1, -1, 2, -2 in the 12 indeterminates x_ijk where i,j = 1,2 and k = 1,2,3. This invariant can be regarded as a natural generalization of Cayley's hyperdeterminant for 2 x 2 x 2 arrays.