About alternating tensors in Michael Spivak's "Calculus on manifolds"

Question

Is this equation obvious or not?

By the way, my proof is here:

Answer 1

The underlined expression is the very definition of $${\rm Alt}(T)(v_1,\ldots,v_\color{blue}{j},\ldots,v_\color{blue}{i},\ldots,v_k).$$You want to manipulate that expression so that $${\rm Alt}(T)(v_1,\ldots,v_\color{green}{i},\ldots,v_\color{green}{j},\ldots,v_k)$$shows up. The way to do this is to make a "change of variables" in the summation, that is, considering $\tau = (i,j)\in S_k$ and using that $S_k\cdot \tau = S_k$, which is exactly what Spivak does in what follows. $$\begin{align}\sum_{\sigma \in S_k} {\rm sgn}(\sigma) T(v_{\sigma(1)}, \ldots, v_{\sigma(\color{blue}{j})},\ldots,v_{\sigma(\color{blue}{i})},& \ldots, v_{\sigma(k)}) = \sum_{\sigma \in S_k} {\rm sgn}(\sigma) T(v_{\sigma(1)}, \ldots, v_{\sigma(\tau(\color{green}{i}))},\ldots,v_{\sigma(\tau(\color{green}{j}))}, \ldots, v_{\sigma(k)}) \\ &\stackrel{(1)}{=}\sum_{\sigma' \in S_k\cdot \tau} {\rm sgn}(\sigma ' \circ \tau^{-1}) T(v_{\sigma'\circ \tau^{-1}(1)}, \ldots, v_{\sigma'(\color{green}{i})},\ldots,v_{\sigma'(\color{green}{j})}, \ldots, v_{\sigma'\circ \tau^{-1}(k)})\\ &\stackrel{(2)}{=} \sum_{\sigma' \in S_k}{\rm sgn}(\sigma' \circ \tau)T(v_{\sigma'(1)}, \ldots, v_{\sigma'(\color{green}{i})},\ldots,v_{\sigma'(\color{green}{j})}, \ldots, v_{\sigma'(k)}) \\ &\stackrel{(3)}{=} - \sum_{\sigma'\in S_k} {\rm sgn}(\sigma')T(v_{\sigma'(1)}, \ldots, v_{\sigma'(\color{green}{i})},\ldots,v_{\sigma'(\color{green}{j})}, \ldots, v_{\sigma'(k)}),\end{align}$$where in $(1)$ we call $\sigma' = \sigma \circ \tau$, note that if $\sigma$ runs through $S_k$, then $\sigma'$ runs through $S_k \cdot \tau$ (which thankfully is $S_k$ again), and substitute everything to get rid of $\sigma$. In $(2)$ we use that $\tau = \tau^{-1}$ fixed everything different from $i$ and $j$. In $(3)$ we use that ${\rm sgn}(\sigma'\circ \tau) = -{\rm sgn}(\sigma')$.