My question is related to page 4 of http://math.uchicago.edu/~may/REU2020/REUPapers/Graham.pdf
Let $e_j=\sum \limits_{1\leq i_1<...<i_j \leq n}x_{i_1}\dots x_{i_j}$ be the elementary symmetric polynomials.
It's $e_0=1$ and $e_\lambda=e_\lambda{_{1}}\dots e_\lambda{_n}$ for a partition $\lambda$.
I want to compute all polynomials of the form $e_\lambda$ in $\mathbb{C}[x_1,x_2,x_3]^{S_3}$ which are of degree $3$.
My way was:
It's $|\lambda|=3$, so the partitions are $\lambda=(3), \lambda=(2,1)$ and $\lambda=(1,1,1)$.
Then I computed it for $\lambda=(3)$ and I got:
$e_\lambda=e_3=x_1x_2x_3$
Where is my mistake here?
The solution says it's $e_3=x_1^3+x_2^3+x_3^3$.
How was it computed?
Edit: Wikipedia also says it's $e_3(x_1,x_2,x_3)=x_1x_2x_3$: https://en.wikipedia.org/wiki/Elementary_symmetric_polynomial
You’re totally fine, the mistake was on the part of the author. Indexing by integers you attain
\begin{align} e_{1}(x_{1},x_{2},x_{3}) &= x_{1} + x_{2} + x_{3}, \\ e_{2}(x_{1},x_{2},x_{3}) &= x_{1}x_{2} + x_{1}x_{3} + x_{2}x_{3}, \\ e_{3}(x_{1},x_{2},x_{3}) &= x_{1}x_{2}x_{3}. \end{align}
Then to get your elementary symmetric polynomials indexed by partitions, you just start multiplying
\begin{align} e_{(3)}(x_{1},x_{2},x_{3}) &= x_{1}x_{2}x_{3}, \\ e_{(2,1)}(x_{1},x_{2},x_{3}) &= x_{1}^{2}x_{2} + x_{1}^{2}x_{3} + x_{1}x_{2}^{2} + x_{2}^{2}x_{3} + x_{1}x_{3}^{2} + x_{2}x_{3}^{2} + 3x_{1}x_{2}x_{3}, \\ e_{(1,1,1)}(x_{1},x_{2},x_{3}) &= x_{1}^{3} + x_{2}^{3} + x_{3}^{3} + 3(x_{1}^{2}x_{2} + x_{1}^{2}x_{3} + x_{1}x_{2}^{2} + x_{2}^{2}x_{3} + x_{1}x_{3}^{2} + x_{2}x_{3}^{2}) + 6x_{1}x_{2}x_{3}. \end{align}
Although I'm not sure how useful this exercise is. In general, I prefer to just leave these in the form
\begin{equation} e_{(7,4^{3},2,1^{4})} = e_{7}e_{4}^{3}e_{2}e_{1}^{4}. \end{equation}