The 3 × 3 × 3 Hyperdeterminant as a Polynomial in the Fundamental Invariants for SL3(C) × SL3(C) × SL3(C)

We briefly review previous work on the invariant theory of 3 × 3 × 3 arrays. We then recall how to generate arrays of arbitrary size m1 × · · · × mk with hyperdeterminant 0. Our main result is an explicit formula for the 3 × 3 × 3 hyperdeterminant as a polynomial in the fundamental invariants I6, I9, I12 for the action of the Lie group SL3(C)× SL3(C)× SL3(C). We apply our calculations to Nurmiev’s classification of normal forms for 3× 3× 3 arrays.