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.