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

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 $(S_3\times S_3\times S_2)⋊S_2$. 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 $(S_4\times S_4\times S_2)⋊S_2$.

arXiv comment15 pages