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This is what's in my book:

Let $\Delta$ be a polynomial given by: $\Delta = \prod_{1 \leq i \leq j \leq n} (x_i - x_j)$

For example, when $n=4$:

$\Delta = (x_1 - x_2)(x_1 - x_3)(x_1 - x_4)(x_2 - x_3)(x_2 - x_4)(x_3-x_4)$

For each $\sigma \in S_n$ let $\sigma$ act on $\Delta$ by permuting the variables in the same way it permutes their indices:

$\sigma(\Delta)=\prod_{1 \leq i \leq j \leq n} (x_{\sigma(i)} - x_{\sigma(j)})$.

And then we have the following example for $n=4$ and $\sigma = (1 2 3 4)$:

$\sigma(\Delta)=(x_2 - x_3)(x_2 - x_4)(x_2-x_1)(x_3-x_4)(x_3-x_1)(x_4-x_1)$

What I don't understand is how they get this result. I don't understand how $\sigma$ acts on $\Delta$. I can see that some terms are flipped (for example ($x_2-x_1$)) and that the order has changed, but why? Could somebody help me out?

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We have $\sigma(1) = 2$ and $\sigma(2) = 3$, so the first factor $(x_1-x_2)$ becomes $(x_{\sigma(1)}-x_{\sigma(2)}) = (x_2 - x_3)$. Similarly for the rest. Does that clear things up?

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  • $\begingroup$ Thanks, you're a hero! It's so obvious. $\endgroup$ – Nikki May 17 '16 at 15:25

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