A Rich "isotropic tensor" concept

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Isotropic tensors maintain their components across all coordinate systems transformed under rotation, with only rank-0, rank-2, and rank-3 tensors existing—specifically, a scalar, the Kronecker delta, and the permutation symbol. There are no rank-1 isotropic tensors, or vectors. The enumeration of isotropic tensors leads to a sequence known as Motzkin sum numbers, which follow a specific recurrence relation. A generating function for these numbers produces a polynomial series that highlights their mathematical beauty. The discussion emphasizes the intricate relationship between mathematics and physics, suggesting that the exploration of such concepts can deepen appreciation for the mathematical framework underlying nature.
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My field is physics and I'm very cautious about the "math describing the Nature" attitude, but I can't help admiring the deep richness of mathematics!

The other day, I was checking about isotropic tensors. An isotropic tensor keeps its components in all coordinated systems transformed under rotation. Then, unexpectedly, I came across some beautiful remarks on the topic. Here are some of them.

How many isotropic tensors exist? There are only single rank-0, rank-2, and rank-3 tensors, respectively, a scalar, the Kronecker ##\delta^{ij}##, and the permutation symbol ##\epsilon_{ijk}##. There are no rank-1 isotropic tensors, that is, vectors.

Now, if one attempts to enumerate all the isotropic tensors, starting with the ones above and going to higher ranks, one gets the sequence:
$$1, 0, 1, 1, 3, 6, 15, 36, 91, 232, {\rm etc.} $$
These numbers are called ##{\it Motzkin\,sum\,numbers}## and obey a recurrence relation:
$$a_n = {n-1\over n+1}\left(2a_{n-1} + 3a_{n-2}\right)$$
with ##a(0) =1, a(1) = 0##.

There is also a generating function:
$$G(x) = {1\over 2x}\left(1 -\sqrt{1-3x\over 1+x}\right)$$
that produces the following polynomial series:
$$\sum_{n=0}^\infty a_n\, x^n = 1 + x^2 + x^3 + 3x^4 + 6x^5 + 15x^6 + \cdots$$
No need to draw your attention to the coefficients of the powers! Plain beautiful!
 
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