Is it possible to create a rank 3 tensor that is skew-symmetric in one pair of indices but symmetric in another pair? That is, a tensor whose components satisfy $$S_{ijk}=-S_{jik}=S_{kji}$$ How many free parameters would such a tensor contain if it exists? I would like to like to parameterise this in the most arbitrary way possible, what would this parameterisation look like?
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1The tensorized version of the Levi-Civita symbol has this property. – K.defaoite Mar 06 '25 at 19:31
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Thanks for pointing me to this! I'm not sure I'm following tho, isn't the Levi-Civita tensor still totally skew-symmetric? – Ben94 Mar 07 '25 at 09:29
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I was just following the single equation in your post (which it satisfies). – K.defaoite Mar 07 '25 at 18:37
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Any such tensor necessarily vanishes (assuming the characteristic of the underlying field is different from $2$). Applying the identities repeatedly we arrive at
$$S_{ijk}=-S_{jik}=-S_{kij}=S_{ikj}=S_{jki}=-S_{kji}=-S_{ijk},$$
hence $S_{ijk} = 0$.
pregunton
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