My question comes from Loring Tu's An Introduction to Manifolds Second Edition.
The problem is
Let $f$ be a $k$-tensor on a vector space $V$. Prove that $f$ is alternating if and only if $f(v_1,\ldots,v_k)=0$ whenever two of the vectors $v_1,\ldots,v_k$ are equal.
The definition used for alternating functions is $\sigma f= (\text{sgn }\sigma)f$.
Proving the rightwards implication ("only if") is easy; I'm having trouble with the reverse direction.
I've tried expanding $f$ into its components using the basis $\{\alpha^{i_1}\otimes\cdots\otimes\alpha^{i_k}\}$ but wasn't able to take that anywhere. I was also considering somehow using the basis $\{\alpha^{i_1}\wedge\cdots\wedge\alpha^{i_k}\}$ for alternating functions, but couldn't think of a way to apply it.
I think my biggest hurdle on this problem is how to use the fact about the special case $v_a=v_b,a\ne b$ to prove anything about the cases where we don't have equivalent arguments.
If anyone has any tips or hints on how to progress, it would be greatly appreciated.
Best Answer
Consider any $(v_i)_{i=1 \ldots k}$ and any $1 \leq j<l \leq k$ distinct integers.
Pose $w_j = w_l = v_j + v_l$ and $w_i = v_i$ for all the other values of $i$.
Now, $w_j = w_l$, so $f(w_1,\ldots,w_k)=0$. However, $$f(w_1,\ldots,w_k) = f(v_1,\ldots,v_j, \ldots, v_l, \ldots ,v_k) + f(v_1,\ldots,v_l, \ldots, v_l, \ldots ,v_k) + f(v_1,\ldots,v_j, \ldots, v_j, \ldots ,v_k) +f(v_1,\ldots,v_l ,\ldots, v_j ,\ldots, v_k) = f(v_1,\ldots,v_j ,\ldots, v_l, \ldots ,v_k) + f(v_1,\ldots,v_l, \ldots, v_j ,\ldots, v_k)$$
This shows it if $\sigma = (jl)$, can you finish from here?