2 research outputs found
Cayley's hyperdeterminant: a combinatorial approach via representation theory
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.Comment: 20 page