Cayley's hyperdeterminant: a combinatorial approach via representation theory

TL;DR

This paper presents a combinatorial and representation-theoretic approach to understanding Cayley's hyperdeterminant, demonstrating its role as a generator of invariants for 2x2x2 arrays and extending the method to higher dimensions.

Contribution

It introduces a new combinatorial method using Lie algebra representations to analyze hyperdeterminants and invariants of multidimensional arrays.

Findings

Hyperdeterminant generates all invariants for 2x2x2 arrays.

Representation theory simplifies the invariant computation.

Method extends to general multidimensional arrays.

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.