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I'm reading a book on GR, and it is going over tensors. They say,

When contracting over a pair of upper indices that are symmetric on one tensor, only the symmetric part of the lower indices will contribute; thus, $$X^{(\mu\nu)}Y_{\mu\nu} = X^{(\mu\nu)}Y_{(\mu\nu)}$$

Here, $X^{(\mu\nu)} = \frac{1}{2}\left(X^{\mu\nu} + X^{\nu\mu}\right)$ is the symmetrization of the tensor. I don't follow this step - why do the asymmetric parts of the lower indices not contribute, and if they don't, even then what then allows us to write that equation?

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    $\begingroup$ Have you tried writing it out and relabelling some dummy indices? $\endgroup$ Mar 27, 2021 at 17:14
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    $\begingroup$ Ah, yeah, it turns out I just needed to do some algebra, which I haven't thought of. I am pretty new to tensors, so I haven't built up a good intuition of what to do when working with them yet. I suppose I'll answer the question and leave it up. $\endgroup$
    – Vedvart1
    Mar 27, 2021 at 17:19
  • $\begingroup$ You could also use that $Y_{\mu \nu} = Y_{(\mu \nu)} + Y_{[\mu \nu]}$, then the above relationship is more intuitive since the product of the symmetric part and the antisymmetric part is zero due to antisymmetry. $\endgroup$
    – Triatticus
    Mar 27, 2021 at 19:31

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Turns out after some algebra,

$$ \begin{align*} X^{(\mu\nu)}Y_{(\mu\nu)} &= \frac{1}{4}\left(X^{\mu\nu}Y_{\mu\nu} + X^{\mu\nu}Y_{\nu\mu} + X^{\nu\mu}Y_{\mu\nu} + X^{\nu\mu}Y_{\nu\mu}\right) \\ &= \frac{1}{4}\left(2X^{\mu\nu}Y_{\mu\nu} + 2X^{\nu\mu}Y_{\mu\nu}\right) & \text{Renaming dummy variables} \\ &= \frac{1}{2}\left(X^{\mu\nu} + X^{\nu\mu}\right)Y_{\mu\nu} \\ &= X^{(\mu\nu)}Y_{\mu\nu} \end{align*} $$

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