The 3 × 3 × 3 Hyperdeterminant as a Polynomial in the Fundamental Invariants for {{SL_3(\mathbb{C}) \times SL_3(\mathbb{C}) \times SL_3(\mathbb{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 \({m_1 \times \cdots \times m_k}\) 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 \({SL_3(\mathbb{C}) \times SL_3(\mathbb{C}) \times SL_3(\mathbb{C})}\). We apply our calculations to Nurmiev’s classification of normal forms for 3 × 3 × 3 arrays.