Confusion with notation of square bracket and round bracket of indices of a tensor

Question

Refer to the following picture: I am confused with the last notation. Say if I got a tensor ${T^{abc}}_{de}$ and I would like to denote a new tensor which is defined by permuting the indices $a$ and $c$. But you just can't add an open bracket before $a$ and a closed bracket after $c$, because in the notation, it means permuting $a,b,c$ instead of just $a,c$ only.

So how should I do? Thank you.

Answer 1

Suppose, $t^{abc}_{de}=U^{ac}⊗V^{b}_{de}$ with $U∈⊗^{2}T_{x}$, $V∈⊗T_{x}⊗^{2}T^{*}_{x}$ with $x∈M$, $M$ some underlying manifold. Then, $\widetilde{t}≑ U^{(ac)}⊗V^{b}_{de}$ and, $\overline{t}≑ U^{[ac]}⊗V^{b}_{de}$.