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I can write out the symmetric sum of for example $x^1_1x_2^2x_3^3$ but I don't understand the notion of $\sum_{sym}x{^1}_{\sigma(1)}x{^2}_{\sigma (2)}x{^3}_{\sigma(3)}$ where $\sigma$ runs over the permutation {1,2,3}. Specifically I don't understand how the sigma function explains the symmetric sum. Is it even a function? And if I were to write out the sum with the $\sigma$ how would it look like?

per persson
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1 Answers1

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The $\sigma$ runs over all possible permutations. So $\sigma$ becomes the following six functions:

$\{1,2,3\} \mapsto \{1,2,3\}$

$\{1,2,3\} \mapsto \{1,3,2\}$

$\{1,2,3\} \mapsto \{2,1,3\}$

$\{1,2,3\} \mapsto \{2,3,1\}$

$\{1,2,3\} \mapsto \{3,1,2\}$

$\{1,2,3\} \mapsto \{3,2,1\}$.

For each function there is a term in the symmetric sum:

$x_1^1x_2^2x_3^3$

$x_1^1x_3^2x_2^3$

$x_2^1x_1^2x_3^3$

$x_2^1x_3^2x_1^3$

$x_3^1x_1^2x_2^3$

$x_3^1x_2^2x_1^3$.

So there are six permutations of $\{1,2,3\}$, each one of them is a function $\sigma$ and for each such function $\sigma$ there is a term in the symmetric sum, namely $x^1_{\sigma(1)}x^2_{\sigma(2)}x^3_{\sigma(3)}$.

Mentastin
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