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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

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