I Explanation of the Alternator

14
0
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

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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.
 
14
0
Thank you very much for your answer, I got it.
 

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