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I was wondering, what kinds of tensors (in the sense of "multi-index arrays of numbers") exist that have conventional names, which can help describing results of tensor calculations.

I can only think of two examples:

  • The 2nd-rank Kronecker delta defined as $$ \delta_{a,b} = \begin{cases} 1 & \mathrm{if}~a=b \\ 0 & \mathrm{otherwise} \end{cases} $$

  • The arbitrary-rank Levi-Civita symbol defined as $$ \epsilon_{a_1,...,a_r} = \begin{cases} +1&\text{if }(a_1,...,a_r)\text{ an even permutation of }(1,...,r)\\ -1&\text{if }(a_1,...,a_r)\text{ an odd permutation of }(1,...,r)\\ 0&\text{otherwise} \end{cases} $$

Purpose

I am trying to find conventional names to express tensors I encountered in group theory. An example was a $3\times3\times3$ tensor $A_{abc}$ with 6 equal elements at the positions where $a,b,c$ are mutually different, which can be expressed as $A_{abc} \propto |\epsilon_{abc}|$, and a $3\times3\times3\times3$ tensor which had 9 equal non-zero elements at positions matching an index pattern $aabb$ (6 elements) or $aaaa$ (3 elemets) (assuming $a\neq b$), which can be written as $A_{abcd} \propto \delta_{ab} \delta_{cd}$.

However, I also came across a tensor which had 21 equal non-zero elements for the patterns $aaaa$ (3), $aabb$ (6), (as above) but additionally also for $abab$ (6) and $abba$ (6). I couldn't figure out any convenient way to express this tensor, that didn't require introducing a new "tensor of ones". The closest I got was $\delta_{ab}\delta_{cd}+\delta_{ac}\delta_{bd}+\delta_{ad}\delta_{bc} - 2\delta_{ab}\delta_{bc}\delta_{cd}$, but that is hardly concise nor is it obvious from looking at the equation that all non-zero components are the same.

This question may be related to asking for tensors which are invariant under orthonormal basis transformations.

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  • $\begingroup$ I haven't seen the notation $\delta_{abcd}$ for the quantity that is $1$ for $a = b = c = d$ and zero otherwise, but one can just write $\delta_{abcd}$ as $\delta_{ab} \delta_{ac} \delta_{ad}.$ $\endgroup$ – Travis May 4 '17 at 13:59
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    $\begingroup$ IIRC the only tensors which are invariant under orthogonal transformations (i.e., elements of $SO(n)$) in a general linear vector space are $\delta_{ab}$, $\epsilon_{abc\cdots}$, and tensor products thereof. $\endgroup$ – Michael Seifert May 4 '17 at 14:10
  • $\begingroup$ If you're looking for a good exposition of invariant subspaces of tensor spaces under $SO(3)$ transformations, I would suggest looking in Hamermesh's Group Theory and its Applications to Physical Problems. Unfortunately I lent out my copy and don't have the time to reconstruct all of the results necessary, or I'd summarize it as an answer. $\endgroup$ – Michael Seifert May 4 '17 at 14:30
  • $\begingroup$ @Travis Good point, I'll update that part. $\endgroup$ – kdb May 4 '17 at 16:44
  • $\begingroup$ You could replace the expression in question with $\delta_{ab} \delta_{ac} \delta_{ad} \delta_{bc} \delta_{bd} \delta_{bc}$. It's ugly and not very parsimonious, but at least it makes it more or less apparently that the resulting expression is symmetric in $a, b, c, d$. $\endgroup$ – Travis May 5 '17 at 10:25

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