The following problem is from "Analysis I" by Amann/Escher.

Exercise: There are obvious operations of $S_m$ on $\mathbb{N}^m$ and on $R[X_1,\dots,X_m]$. A polynomial $p\in R[X_1,\dots,X_m]$ is called symmetric if $S_mp=\{p\}$, i.e. if it is fixed by every permutation. Show the following ($X$ is defined as the tuple $(X_1,\dots,X_m)$ ):

  1. A polynomial $p$ is symmetric if and only if it has the form $$p=\sum_{[\alpha]\in\mathbb{N}^m/S_m}a_{[\alpha]}\left(\sum_{\beta\in[\alpha]}X^\beta\right)\,.$$
  2. The elementary symmetric functions $$s_k:=\sum_{1\le j_1\le\dots\le j_k\le m}X_{j_1}\cdot\dots{}\cdot X_{j_k}$$ are symmetric.
  3. Consider the polynomial $$p=(Y-X_1)(Y-X_2)\cdot\dots{}\cdot(Y-X_m)\in R[X_1,\dots,X_m][Y]\,.$$ Show that $$p=\sum_{k=0}^m(-1)^ks_kY^{m-k}\,,$$ where $s_0:=1$.

My attempt:

  1. From general facts about group actions I know that $\mathbb{N}^m$ is the disjoint union of orbits $[\alpha]=S_m\alpha$, so I can always write $$p=\sum_{\alpha\in\mathbb{N}^m}a_\alpha X^\alpha=\sum_{[\alpha]\in\mathbb{N}/S_m}\sum_{\beta\in[\alpha]}a_\beta X^\beta\,.$$ Furthermore, applying a permutation $\sigma$, I get $$\sigma p=\sum_{[\alpha]\in\mathbb{N}/S_m}\sigma\left(\sum_{\beta\in[\alpha]}a_\beta X^\beta\right)\,.$$ A term of the form $\sigma\left(\sum_{\beta\in[\alpha]}a_\beta X^\beta\right)$ will again be of the form $\sum_{\gamma\in[\alpha]}a_\gamma X^\gamma$, because $\sigma[\alpha]=[\alpha]$. (This is the step I am most insecure about.) Thus $p$ is symmetric if and only if for every $\alpha$ the term $\sum_{\beta\in[\alpha]}a_\beta X^\beta$ is symmetric, which is the case if and only if all the $a_\beta$ coincide. QED
  2. I think I can rewrite $$s_k=\sum_{\begin{matrix}J\subseteq\{1,\dots,m\}\\\#J=k\end{matrix}}\prod_{j\in J}X_j\,.$$ The claim then follows, because every permutation of $\{1,\dots,m\}$ induces a permutation of the set of all subsets of $\{1,\dots,m\}$ with cardinality $k$. Is this correct?
  3. I thought about induction on $m$, but I am not succesful. I hope you can help me.

I am particularly interested in wether my ideas concerning 1. are written down precise enough. Thank you in advance.

  • $\begingroup$ 2. follows immediately from 3. and an inductive proof of 3. can be found here. (I probably could answer to 1., too if I'd understand your notation.) $\endgroup$ – user26857 Dec 29 '14 at 21:30
  • $\begingroup$ @user26857 What exactly is unclear about 1.? Given $\alpha\in\mathbb{N}^m$, $[\alpha]$ denotes the orbit of $\alpha$ under the action of $S_m$. $\mathbb{N}^m/S_m$ denotes the set of all such orbites. Thank you for the link, I'll have a look at it. Is my solution of 2. correct? Thank you. $\endgroup$ – user114885 Dec 30 '14 at 11:43

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