Exercise 2.2.7

Proof. We use the relations (2.31) p. 48, (or (6.7) p. 138) proved in Exercises 6.4.3 and 6.4.4 : for all $\sigma ,\tau \in S_n$, and all $f,g\in F[x_1,x_2,\cdots ,x_n]$ :

(We will use the notation $\tau \circ \sigma =\mathit{\tau \sigma }$.)

Let $g={\prod <!--\; FUNCTION\; APPLICATION\; -->}_{\sigma \in S_n}\sigma \cdot f$.

Then, if $\tau \in S_n$, using (2) and (3)

As the application $S_n\to S_n,\sigma \mapsto \mathit{\tau \sigma }$ is bijective, the index change ${\sigma }^{\prime }=\mathit{\tau \sigma }$ gives

So, for all $\tau \in S_n,\tau \cdot g=g$ : thus $g$ is a symmetric polynomial.

Same proof for $\tau .{\sum <!--\; FUNCTION\; APPLICATION\; -->}_{\sigma \in S_n}\sigma \cdot f={\sum }_{\sigma \in S_n}\sigma \cdot f$ : use (2) in place of (3).

Conclusion : $\underset{\sigma \in S_n}{\prod <!--\; FUNCTION\; APPLICATION\; -->}\sigma \cdot f$ and $\underset{\sigma \in S_n}{\sum <!--\; FUNCTION\; APPLICATION\; -->}\sigma \cdot f$ are symmetric polynomials. □