When I first became interested in hyperdeterminants about ten years ago, there were very few people researching them. Recently they have emerged as interesting in physics, expecially quantum entanglement and superstring theory. This has led to a resurgence of interest in hyperdeterminants.

Of course the work of Gelfand, Kapranov, and Zelevinsky who wrote the hyperdeterminant book led the way. Another longterm practitioner of the art is David Glynn who has done important work on multiplicative hyperdeterminants and the relationship with coding theory. He is probably the leading expert on much of the stuff that I have started to get interested in. David recently left some comments on my blog article about Cayley’s Hyperdeterminants so I think this is a good moment to continue this blog with some posts in which I will try to understand and followup on his comments.

David drew my attention to a different type of hyperdeterminant which Cayley first found in 1843 two years before he defined the hyperdeterminant as a discriminant that I use here. This determinant for a hypermatrix A of size m^{n} is written det_{0}(A) to distinguish it from the discriminant version. det_{0}(A) is a polynomial of degree m in the components of A and it is an SL(m)^{n} invariant when n is even. In that case it can be expressed by contracting the nm indices of m copies of the n indices of the hypermatrix over the mn indices of n copies of the m indices of the Levi-Civita symbol. When n is odd this construction degenerates to zero because of the conflict between the symmetry of the product of hypermatices and the anti-symmetry of the product of Levi-Civita symbols, but when n is even it gives an invariant that is det_{0}(A) up to a normalisation. When n=2 this gives the familiar determinant of a square matrix but for n > 2 it is different from the hyperdeterminant of hypermatrices defined as discriminants.

That much was already familiar to me, but David also remarked that this hyperdeterminant has a multiplicative property that generalises the well known property of matrices i.e. det(AB) = det(A)det(B). This was something I had not really appreciated before. The product of two hypermatrices is taken to be the contraction of an index from one hypermatrix with an index from the other. E.g. the product of an m^{n} hypermatrix with an m^{k} hypermatrix is an m^{n+k-2 }hyprmatrix. The discriminant hypermatrix has a similar multiplicative property for boundary format hypermatrices but not for more general formats. One other case where a multiplicative property works is the product of a hypermatrix with a matrix as I demonstrated when looking at the hypermatrix as an invariant.

Looking at how all these cases work I now realise that they are all special cases of a general result. Suppose A and B are two hypermatrices that may be of different formats but they have a matching pair of indices so that a product AB can be formed by contracting over those indices. Suppose also that there are invariants for each format I_{1}(A) and I_{2}(B) that have the same polynomial degree d. Then there is always a third invariant I_{3}(C) of degree d on the format of the product hypermatrix which satisfies a multiplicative property I_{3}(AB) = I_{1}(A) I_{2}(B). If we have two invariants I_{1} and I_{2} of different degrees d1 and d2 we can of course form invariants of degree d = lcm(d1,d2) (least common multiple of d1 and d2), by taking powers of I_{1} and I_{2} then the result applies to those.

Before I show why this result is true let’s look at how it works for the special cases. if A and B are of size m^{n} and m^{k} then det_{0}(A) and det_{0}(B) are both of degree m. So there must be an invariant of degree m such that I(AB) = det_{0}(A) det_{0}(B) , but the only invariant of the required degree is det_{0}(AB) up to a factor so det_{0}(AB) = f det_{0}(A) det_{0}(B) for a suitable factor f independent of A and B. The factor f can be shown to be one for a suitable normalisation in the definition of det_{0 }by checking one test case. A similar argument works for the case of hyperdeterminants of boundary format hypermatrices when the contraction indices are choosen so that the product of two boundary hypermatrices is another boundary hypermatrix and exponents are used to construct invariants of the same degree. This follows provided the hyperdeterminants are unique invariants of the specific degree for boundary format hypermatrices, which I have not checked. Other products of hypermatrices raised to suitable exponents will give some invariant on the product. It will not in general be a hyperdeterminant but its form can be worked out in specific cases.

So why is this general multiplicative property of invariants true? It follows from the fact that all hypermatrix invariants can be constructed using contractions over sums and products of the Levi-Civita symbols ε^{a…c} together with a general identity which reduces the product of two such symbols to an antisymmetric sum over products of the Kronecker delta symbol.

ε^{a…c} ε^{d…f} = δ^{a [d} … δ^{c f] }

(where the square brackets indicate a normalised antisymmetrization over all the second indices of the deltas)

Express the product of invariants I_{1}(A) and I_{2}(B) using Levi-Civita symbols then plug in this identity into the expression for all pairs of symbols over the two indices subject to contraction. The deltas then form combinations of products of the hypermatrices and the result is obtained.