What's are these index objects called? And $\mathrm{\LaTeX}$ \sum question

I want to refer to $$A_iB_jC_k$$

using $$\psi(ijk) = A_iB_jC_k$$

So that I can write out quite overwhelming-looking sums of ABC terms as sums of terms that look like 123, 231, 113, etc. If I am not treating "231" as an integer but a term in "index space", what is the proper name given to such terms?

For instance, if we let $\psi : ijk \rightarrow \hat{i}A_jB_k$ be the linear map taking terms in index space $ijk$ into "$AB$ sum space" (your normal notation space), then we can write

$$(1) \;\;\;\;\;\; \psi^{-1}(\bar{A} \times \bar{B}) = \sum i(jk - kj) = \sum k(ij - ji) = \sum j(ki - ik)$$

where $\bar{A}, \bar{B} \in \mathbb{R}^3$, where the sum is taken over all even permutations of 123 (ie 123, 312, 231). In other words, the cross-product in 3 dimensions has those index space formulas.

Why do this? Because when we deal with proving trickier vector identities like one for $(A\times B)\times(C\times D)$, writing down the letters with each index takes way to long when doing it by hand. Also, some of the identity proofs hinge on equivalences in index space and to see the equivalence clearer by taking out irrelevant symbols is better.

(ANSWERED ALREADY) LaTex Question: in (1) how would I write that sum? My first guess was over $A_3$ or the set of even perms on (1,2,3). Brevity is the aim. And where can I find how to do this on my own (I want $A_3$ to be on the bottom of the sigma)? A Thanks.

I've come up with a couple of neat proofs using this technique. If anyone is interested I could post them and a comparison to the regular-style proof of vector identities.

EDIT: I've answered my LaTeX question. I would simply use _ and ^: \sum_{A_3}^{hi, there!} to get:

$$\sum_{A_3}^{hello}$$

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1 Answer

I'd call $(i,j,k)$ a "multi-index".

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What about ijk? The two parentheses and commas take considerably longer to write out. – Enjoys Math Mar 8 '12 at 3:50