On the hyperdeterminant for 2×2×3 arrays

Abstract

We use the representation theory of Lie algebras and computational linear algebra to determine the simplest non-constant invariant polynomial in the entries of a general 2 × 2 × 3 array. This polynomial is homogeneous of degree 6 and has 66 terms with coefficients ±1, ±2 in the 12 indeterminates x _ijk where i, j = 1, 2 and k = 1, 2, 3. This invariant can be regarded as a natural generalization of Cayley's hyperdeterminant for 2 × 2 × 2 arrays.

Keywords:

• hyperdeterminant
• multilinear algebra
• representation theory
• invariant theory
• multidimensional arrays

AMS Subject Classifications::

• Primary
• 15A69
• Secondary
• 15A15
• 17B10

Acknowledgements

The research was partially supported by a Discovery Grant from NSERC, the Natural Sciences and Engineering Research Council of Canada.