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)

2013/10/11 by Bremner, Murray, Hu, Jiaxiong, Oeding, Luke
#17B10 #Algebraic Geometry (math.AG) #FOS: Computer and information sciences #FOS: Mathematics #Primary 13A50 #Representation Theory (math.RT) #Secondary 15A72 #Symbolic Computation (cs.SC)

paper · doi:10.48550/arxiv.1310.3257

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 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.

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