Hyperdeterminant information

In algebra, the hyperdeterminant is a generalization of the determinant. Whereas a determinant is a scalar valued function defined on an n × n square matrix, a hyperdeterminant is defined on a multidimensional array of numbers or tensor. Like a determinant, the hyperdeterminant is a homogeneous polynomial with integer coefficients in the components of the tensor. Many other properties of determinants generalize in some way to hyperdeterminants, but unlike a determinant, the hyperdeterminant does not have a simple geometric interpretation in terms of volumes.

There are at least three definitions of hyperdeterminant. The first was discovered by Arthur Cayley in 1843 presented to the Cambridge Philosophical Society.^[1] It is in two parts and Cayley's first hyperdeterminant is covered in the second part.^[1] It is usually denoted by det₀. The second Cayley hyperdeterminant originated in 1845^[2] and is often denoted "Det". This definition is a discriminant for a singular point on a scalar valued multilinear map.^[2]

Cayley's first hyperdeterminant is defined only for hypercubes having an even number of dimensions (although variations exist in odd dimensions). Cayley's second hyperdeterminant is defined for a restricted range of hypermatrix formats (including the hypercubes of any dimensions). The third hyperdeterminant, most recently defined by Glynn, occurs only for fields of prime characteristic p. It is denoted by det_p and acts on all hypercubes over such a field.^[3]

Only the first and third hyperdeterminants are "multiplicative," except for the second hyperdeterminant in the case of "boundary" formats. The first and third hyperdeterminants also have closed formulae as polynomials and therefore their degrees are known, whereas the second one does not appear to have a closed formula or degree in all cases that are known.

The notation for determinants can be extended to hyperdeterminants without change or ambiguity. Hence the hyperdeterminant of a hypermatrix A may be written using the vertical bar notation as |A| or as det(A).

A standard modern textbook on Cayley's second hyperdeterminant Det (as well as many other results) is "Discriminants, Resultants and Multidimensional Determinants" by Gel'fand, Kapranov and Zelevinsky.^[4] Their notation and terminology is followed in the next section.