Question About Indices of Alternating Tensors

Question

So, with an alternating $k$-tensor, $T^{\sigma} = (-1)^{\sigma}T$. But does that mean $T^{\sigma}(v_1, \cdots, v_k) = (-1)^{\sigma}T(v_1, \cdots, v_k)$ or does it mean $T^{\sigma}(v_1, \cdots, v_k) = (-1)^{\sigma}T(v_{\sigma^{-1}(1)}, \cdots, v_{\sigma^{-1}(k)})$. I have not been able to find a clear answer in my textbook or looking online, and want to make sure I'm understanding correctly. Thanks!

Answer 1

Allow me to summarise the discussion in the comments.

Suppose $T$ is a $(0, k)$-tensor, then for any $\sigma \in S_k$ we define a new $(0, k)$-tensor $T^{\sigma}$ by $T^{\sigma}(v_1, \dots, v_k) := T(v_{\sigma(1)}, \dots, v_{\sigma(k)})$. If $T$ is alternating, then we have

$$T^{\sigma}(v_1, \dots, v_k) = T(v_{\sigma(1)}, \dots, v_{\sigma(k)}) = (-1)^{\operatorname{sign}(\sigma)}T(v_1, \dots, v_k)$$

so $T^{\sigma} = (-1)^{\operatorname{sign}(\sigma)}T$. In particular, if $\sigma$ is an even permutation, then $T^{\sigma} = T$, and if $T$ is an odd permutation, then $T^{\sigma} = -T$.