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I was reading a paper and I came across some notation that confused me. This was concerning the extrinsic curvature tensor defined as $K_{ab} = \frac{\mu}{2}K^i_{(a}K^i_{b)}$. I am normally comfortable with tensor indices, but the placement of the brackets seemed strange. This was not a typo, as later on, similar notation was used for another tensor $\mathcal{K}_{ab} = \mu^2K^i_{[a}E^i_{b]}$. Is this a standard notation used in these areas, and what does it refer to?

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This is indeed not a typo: the round brackets mean symmetrization over the enclosed indices, the square ones mean anti-symmetrization. This notation is not specific to LQG/geometrodynamics, but is a standard one (as acknowledged by the linked wikipedia page on antisymmetric tensors).

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Yes: parentheses and brackets are standard notation. They mean symmetrisation and anti-symmetrisation: $$ \begin{aligned} A_{(\mu\nu)}\equiv\frac{1}{2}(A_{\mu\nu}+A_{\nu\mu})\\ A_{[\mu\nu]}\equiv\frac{1}{2}(A_{\mu\nu}-A_{\nu\mu}) \end{aligned} $$ and similar relations for higher-order tensors. One should note that the factor $\frac12$ is sometimes omitted by some authors.

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