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 m1 x ... x mk 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.
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