Multiple cross products

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In mathematics, there are tricks for multiple cross products. The cross product operation is not associative: we have in general

(A×B)×C ≠ A×(B×C).

On the other hand exterior algebra is associative. Therefore this is an aspect of multilinear algebra that goes beyond the basics of the wedge product. Since the cross product is anticommutative, left and right can be switched with a change of sign, at most.

In traditional treatments of tensors, this question is handled in terms of the Levi-Civita symbol defined by

\epsilon_{ijk} = \left\{ \begin{matrix} +1 & \mbox{if } (i,j,k) \mbox{ is } (1,2,3), (2,3,1) \mbox{ or } (3,1,2)\\ -1 & \mbox{if } (i,j,k) \mbox{ is } (3,2,1), (1,3,2) \mbox{ or } (2,1,3)\\ 0 & \mbox{otherwise: }i=j \mbox{ or } j=k \mbox{ or } k=i \end{matrix} \right.

and a basic identity for it.

Since the cross product as cartesian tensor is

εijkaibj

with the summation convention understood, the required identity would be for

εijkεklm.

This is shown to be a combination of Kronecker deltas

δilδjm − δimδjl.

This can be proved by a short direct argument on permutations; it is also equivalent to an identity on triple cross products.

Armed with this formula, any multiple cross product can be simplified. Those of odd length come out without cross products, since an even number of ε symbols will always 'cancel' into δ symbols. For long products the result does grow exponentially.

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