Exhibit an action of Sym(4) on a set of polynomials

Let n be a positive integer and let R be the set of all polynomials with integer coefficients in the independent variables x_1, x_2, \ldots, x_n. I.e., elements of R are formal sums \sum_{i \in I} a_i \prod_{j=1}^n x_i^{r_{i,j}} for some finite set I, integers a_i, and nonnegative integers r_{i,j}.

For each \sigma \in S_n, define a mapping \sigma \cdot by \sigma \cdot \sum_{i \in I} a_i \prod_{j=1}^n x_j^{r_{i,j}} = \sum_{i \in I} a_i \prod_{j=1}^n x_{\sigma(j)}^{r_{i,j}}.

Prove that this defined a left group action of S_4 on R.

Let p = \sum_{i \in I} a_i \prod_{j=1}^n x_j^{r_{i,j}}. We have 1 \cdot p = 1 \cdot \sum_{i \in I} a_i \prod_{j=1}^n x_j^{r_{i,j}} = \sum_{i \in I} a_i \prod_{j=1}^n x_{\mathsf{id}(j)}^{r_{i,j}} = \sum_{i \in I} a_i \prod_{j=1}^n x_j^{r_{i,j}} = p.

Now let \sigma, \tau \in S_n. Then

\sigma \cdot (\tau \cdot p)  =  \sigma \cdot (\tau \cdot \sum_{i \in I} a_i \prod_{j=1}^n x_j^{r_{i,j}})
 =  \sigma \cdot \sum_{i \in I} a_i \prod_{j=1}^n x_{\tau(j)}^{r_{i,j}}
 =  \sum_{i \in I} a_i \prod_{j=1}^n x_{\sigma(\tau(j))}^{r_{i,j}}
 =  \sum_{i \in I} a_i \prod_{j=1}^n x_{(\sigma \circ \tau)(j)}^{r_{i,j}}
 =  (\sigma \circ \tau) \cdot \sum_{i \in I} a_i \prod_{j=1}^n x_j^{r_{i,j}}

Thus we have a group action of S_n on R.

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