Why is the Alternator Alternating?

[Maxi1995]

Hello,
let us define the Alternator $$Alt(T)$$ where T is a multilinear function
$$Alt(T):= \frac{1}{k!} \sum_{\sigma \in S_n} sgn(\sigma) T (v_{\sigma(1)},...,v_{\sigma(k)}))$$.

Further recognize that

$$S_n$$

is the group of permutations and sgn the signum of the permutation.
May someone explain me why the alternator is alternating, thus to say if I change two argument vectors in Alt, then a minus will appear, as an example

$$w(v_1,...,v_n)=- w(v_n,...,v_1)$$

 

[fresh_42]

It should be formulated a bit more precise. Let's assume we have a tensor $T=v_1\otimes \ldots \otimes v_k$.
Then the alternator makes an alternating tensor out of it, that is it is a mapping from non-alternating to alternating tensors, because it counts the number of mismatches ($\tau = (12)$):
$$
\operatorname{Alt}_2(T)=\operatorname{Alt}_2(v_1\otimes v_2)= \dfrac{1}{2} \left(\operatorname{sgn}(\operatorname{id}) v_{\operatorname{id}(1)}\otimes v_{\operatorname{id}(2)} + \operatorname{sgn}(\tau)v_{\tau(1)}\otimes v_{\tau(2)} \right)=\dfrac{1}{2}\left(v_1\otimes v_2 - v_2\otimes v_1\right)
$$
If you now look on what $\operatorname{Alt}_2$ did with $T=v_1\otimes v_2$, you will find $\operatorname{Alt}_2(v_1\otimes v_2)=-\operatorname{Alt}_2(v_2\otimes v_1)$ which is why it is called alternator. This generalizes to $k-$homogenous tensors and by replacing $\dfrac{1}{k!}$ by $\binom{k+l}{k}$ to pairs of tensors of different ranks.

 

[Maxi1995]

Thank you very much for your answer, I got it.