An Algebraic Duality

Consider a four-dimensional array A whose elements are denoted by the subscripted symbol A_abcd where the indices a,b,c,d can each take on any integer value from 1 to N. Thus the array consists of N⁴ values. In many applications we find that the elements of an array possess symmetries and/or anti-symmetries, which greatly reduce the number of independent elements in the array. For example, if the general array element A_abcd is defined as (a-b)(c-d) then obviously transposing the first two indices has the effect of negating the sign, as does transposing the last two indices. Also, swapping the first and second pairs of indices leaves the value unchanged. This set of symmetries arises in many different contexts. We can express them in the form

In addition, because of the identity

we also often have the skew symmetry

Given a four-dimensional array A with all these symmetries, let us define another array B as follows:

From this we can directly verify the symmetries

Notice that transposing the first two (or the last two) indices leaves the B value unchanged, so the symmetries of the B array are not identical to those of the A array. One is symmetric and the other is anti-symmetric under transposition of the first two (or last two) indices. Nevertheless, the formal properties of these two arrays are quite similar. We can also directly verify that the B array satisfies the skew symmetry

just as does the A array. Furthermore, from these symmetries it follows that the relation (1) is invertible, i.e., we have

This can be shown by direct substitution using equation (1), which leads to

Making use of the transposition symmetries of A, we see that the two main diagonals contribute A_abcd/3 and A_dcba/3, and the outer terms of the inner rows contribute –A_cabd/3, and the inner terms of the outer rows contribute –A_bcad/3. Again making use of the transposition symmetries, we see that the second contribution equals the first, and we can re-arrange the indices of the third and fourth contributions so that the leading index is a. This gives

If we now add and subtract Aabcd/3 to the right hand side we get

But the quantity in parentheses on the right side vanishes due to the skew symmetry of A, so we have the result (2).

This duality arises (for example) in differential geometry, in the relation between the components R_abcd of the covariant curvature tensor (at the origin of Riemann normal coordinates) and the second partial derivatives g_ab,cd of the metric tensor. The skew symmetry of the curvature tensor leads directly on differentiation to the Bianchi identities, which on contraction give the gravitational field equations of general relativity.

Several conditions must be met in order for a duality like this to exist. First, the basic array must have the required symmetries. Second, the indices on the right sides of relations (1) and (2) must each be just a single transposition from the indices on the left sides. Since a transposition has order 2, these relations are “invertible” in the sense that if we make a list of the four permutations of A that contribute to each of the 24 permutations of B, and vice versa, they are the same lists. The mapping makes use of only four of the six simple transpositions. Specifically, we apply the index transpositions (2,3), -(3,1), (1,4), and -(4,2), so each index position is affected by precisely two of these permutations, one contributing positively and the other negatively. The next higher order of permutations are the 3-cycles, but if we form the sum of the four terms given by holding each index fixed and rotating the other three, the result is identically null, i.e.,

assuming the other symmetries of A still apply.

We might try to construct higher-order analogies of this kind of duality, possibly based on the symmetries of quantities of the form A_abcdef = (a-b)(c-d)(e-f) for six-dimensional arrays, but it isn’t clear what identity would serve the role of the skew symmetry for quantities of this form. (Holding one index fixed and cycling the remaining indices does not yield a null sum.) Another possible generalization would be to consider mappings based on index permutations of higher order. In other words, instead of the dual mappings A = f(B) and B = f(A) based on index permutations of order 2, we could seek a cycle of arrays with the mappings A = f(B), B = f(C), and C = f(A) based on index permutations of order 3.

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