20
$\begingroup$

I know that rank 2 tensors can be decomposed as such. But I would like to know if this is possible for any rank tensors?

$\endgroup$
21
$\begingroup$

A (higher) $n$-rank tensor $T^{\mu_1\ldots \mu_n}$ with $n\geq 3$ cannot always be decomposed into just a totally symmetric and a totally antisymmetric piece. In general, there will also be components of mixed symmetry.

The symmetric group $S_n$ acts on the indices $$(\mu_1,\ldots ,\mu_n)\quad \longrightarrow\quad (\mu_{\pi(1)},\ldots ,\mu_{\pi(n)})$$ via permutations $\pi\in S_n$. One can decompose the tensor $T^{\mu_1\ldots \mu_n}$ according to irreps (irreducible representations) of the symmetric group.

Each irrep corresponds to a Young tableau of $n$ boxes. For instance, a single horizontal row of $n$ boxes corresponds to a totally symmetric tensor, while a single vertical column of $n$ boxes corresponds to a totally antisymmetric tensor. But there are also other Young tableaux with a (kind of) mixed symmetry.

Here is a Google search for further reading.

$\endgroup$
  • $\begingroup$ Thanks, I always think this way but never really convince. $\endgroup$ – Saksith Jaksri Feb 19 '17 at 11:32
7
$\begingroup$

No.

Definitions:

  • A rank-n tensor is a linear map from n vectors to a scalar.
  • A symmetric tensor is one which in which the order of the arguments doesn't matter.
  • An antisymmetric tensor in which transposing two arguments multiplies the result by -1.

Suppose we have some rank-3 tensor $T$ with symmetric part $S$ and anti-symmetric part $A$ so

$$T(a,b,c) = S(a,b,c) + A(a,b,c)$$

where $a,b,c\,$ are arbitrary vectors. Transposing $c$ and $a$ on the right hand side, then transposing $a$ and $b$, we have

$$T(a,b,c) = S(c,a,b) + A(c,a,b)$$

so we conclude

$$T(a,b,c) = T(c,a,b)$$

it is trivial to construct a counterexample, so not all rank-three tensors can be decomposed into symmetric and anti-symmetric parts.

$\endgroup$
  • 2
    $\begingroup$ Is it possible to find a more general decomposition into tensors with certain symmetry properties under permutation of the input arguments? $\endgroup$ – Lagerbaer Nov 28 '12 at 22:35
  • $\begingroup$ @Lagerbaer see physics.stackexchange.com/q/18228 $\endgroup$ – Mark Eichenlaub Nov 28 '12 at 22:53

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.