The 3 x 3 x 3 hyperdeterminant as a polynomial in the fundamental invariants for SL(3,C) x SL(3,C) x SL(3,C)
Murray Bremner, Jiaxiong Hu, Luke Oeding
TL;DR
This work provides an explicit polynomial expression for the $3 \times 3 \times 3$ hyperdeterminant $\Delta_{333}$ in terms of the fundamental invariants $I_6$, $I_9$, and $I_{12}$ under the action of $SL(3,\mathbb{C})^3$, namely $\Delta_{333} = I_6^3 I_9^2 - I_6^2 I_{12}^2 + 36 I_6 I_9^2 I_{12} + 108 I_9^4 - 32 I_{12}^3$. The authors derive the coefficients via a projective Gaussian-elimination framework, using modular random arrays and rational reconstruction to confirm the exact form, and provide a Maple worksheet as ancillary material. They apply the result to Nurmiev's classification of normal forms for $3 \times 3 \times 3$ arrays, computing explicit invariant values on the five semisimple families and demonstrating how the invariants distinguish orbit closures and nilpotent cases. Additionally, the paper analyzes how the invariants behave on arrays of known rank (up to Kruskal’s bound $r \le 5$), showing systematic vanishing patterns that align with rank, hypersurface structure, and the Segre dual geometry, thereby linking classical invariant theory with modern tensor and orbit-closure classifications.
Abstract
We briefly review previous work on the invariant theory of 3 x 3 x 3 arrays. We then recall how to generate arrays of arbitrary size m_1 x ... x m_k with hyperdeterminant 0. Our main result is an explicit formula for the 3 x 3 x 3 hyperdeterminant as a polynomial in the fundamental invariants of degrees 6, 9 and 12 for the action of the Lie group SL(3,C) x SL(3,C) x SL(3,C). We apply our calculations to Nurmiev's classification of normal forms for 3 x 3 x 3 arrays.