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)

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.

10 pages, to appear in Mathematics in Computer Science (Special Issue on Computational Algebraic Geometry)

References in corpus (1)

• E_6 and the bipartite entanglement of three qutrits

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• Maximally entangled real states and SLOCC invariants: the 3-qutrit case
• Noncommutative resolutions and CICY quotients from a non-abelian GLSM