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Antisymmetric and symmetric part of a general tensor

  1. Mar 7, 2013 #1
    I've seen it stated several times that a general covariant or contravariant tensor of rank n can be separated into it's symmetic and antisymmetric parts

    [tex] T^{\mu_1 \ldots \mu_n} = T^{[\mu_1 \ldots \mu_n]} + T^{(\mu_1 \ldots \mu_n)}[/tex]

    and this is easy to prove for the case n=2, but I don't see how to prove it for a general n. Could anyone help me out?

    (suspect this has been posted in the wrong forum, but I also suspect that the general relativists can answer this question as good or better than the mathematicians)
  2. jcsd
  3. Mar 7, 2013 #2


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    You're right, this statement holds only for two indices. For three or more you can separate the totally symmetric part and the totally antisymmetric part, but there's leftovers. This is easy to confirm just by counting up the independent components of the symmetric and antisymmetric parts and seeing that they don't add up to the expected total. We say the leftover part has "mixed" symmetry.
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