Prove that $\theta: M^m \rightarrow T^mM$ such that$ (x_1,...,x_m) \rightarrow \frac{1}{m!} \sum_{σ∈Sm}ε(σ)x_{σ(1)} ⊗ · · · ⊗ x_{σ(m)}$ is alternating

Question

How to prove that $\theta: M^m \rightarrow T^mM$ defined as $ (x_1,...,x_m) \rightarrow \frac{1}{m!} \sum_{σ∈Sm}ε(σ)x_{σ(1)} ⊗ · · · ⊗ x_{σ(m)}$ is alternating ?

Alternating (for a map) means that as soon as $x_i = x_j$ for $i \neq j$ the map equal $0$.

This is a classical notion of exterior algebra topic on modules, but I'm struggling a lot with this specific map. I guess I'm missing something obvious about permutation trick.

Thank you in advance for any help !

Answer 1

Assume $x_i=x_j$ and let $(i\,j)$ be the transposition that swaps $i$ and $j$.

For any permutation $\sigma\in S_n$, we can assign a pair $(i\,j)\circ\sigma$ (note that $(i\,j)\circ(i,\,j)\circ\sigma=\sigma$) that has the opposite parity as $\sigma$, because $$\varepsilon((i\,j)\sigma)\ =\ \varepsilon((i\,j))\cdot\varepsilon(\sigma)\ =\ -\varepsilon(\sigma)\,.$$

So $\theta(x_1,\dots,x_n)$ indeed becomes $0$.