Cayley's hyperdeterminant: a combinatorial approach via representation theory

Abstract

Cayley's hyperdeterminant is a homogeneous polynomial of degree 4 in the 8 entries of a 2 x 2 x 2 array. It is the simplest (nonconstant) polynomial which is invariant under changes of basis in three directions. We use elementary facts about representations of the 3-dimensional simple Lie algebra sl_2(C) to reduce the problem of finding the invariant polynomials for a 2 x 2 x 2 array to a combinatorial problem on the enumeration of 2 x 2 x 2 arrays with non-negative integer entries. We then apply results from linear algebra to obtain a new proof that Cayley's hyperdeterminant generates all the invariants. In the last section we show how this approach can be applied to general multidimensional arrays.