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I'm trying to decide which of the following are alternating tensors in $\mathbb R^4$ and express those that are in terms of the elementary tensors on R^4:

$f(x,y) = x_1y_2 = x_2y_1 + x_1y_1$ \begin{align*} g(x, y) &= x_1y_3 - x_3y_2 \\ h(x, y) &= x_1^3y_2^3 - x_2^3y_1^3 \end{align*}

All I can tell is that we have to define a $\sigma$ permutation by $f(\sigma(x),\sigma(y) =$

and I'm trying to go about solving it.

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    I tried to clean this up, but the formula for $f$ doesn't make sense as written and the remarks after the display don't seem to be finished. Please edit!2012-03-19
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    It looks like your "tensors" are maps $(\mathbf R^4)^2 \to \mathbf R$ and are functions of two arguments. In that case, if you have shown that one of them is linear in each argument ($h$ looks somewhat suspect), you really only need to check that $f(y, x) = -f(x, y)$ for all $x, y \in \mathbf R^4$. This is just the condition for being an alternating bilinear form, so there's no need to think too hard about permutations.2012-03-19
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    If the second $=$ in the line describing $f$ means to be an $=$, or something else? And the sentence "All I can tell is that we have to define..." is incomplete (and the formula has unbalanced parentheses, not to mention a completely AWOL right hand side of the equality!)2012-03-19
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    I will edit it above2012-03-19
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    Thanks for catching the incompleteness2012-03-19
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    The equation for $f$ still doesn't make much sense to me.2012-03-19

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