What exactly is the symmetric product on the affine line $\mathbb{A}^1$? I am trying to prove some things about $\mathbb{A}^n/S_n$ the symmetric product of the affine line, such as that it is an affine variety or that it is isomorphic to $\mathbb{A}^n$ or that the quotient map into it is an morphism of varieties. But I lack understanding about what exactly this space is or how precisely it is defined. Wikipedia defines it as the orbit space of the group action under $S_n$ defined by $S_n\times \mathbb{A}^n:(\sigma,a)\mapsto \sigma(a)$. But I am unsure what precisely this means. Does it map every element of $\mathbb{A}^n$ to a class of permutations of its coordinates? Surely, in this way it will not be isomorphic to $\mathbb{A}^n$. I know there is also a definition involving tensor products but I am not sure how exactly it is defined in this way.
Can someone give me a good definition of the symmetric product and some intuition with which I can tackle these problems?
 A: The (closed) points of $\Bbb{A}^n_k/S_n$ are the subsets of $\Bbb{A}^n_k$ of the form $\bigcup_{g\in S_n} g(a)$ with $a\in \Bbb{A}^n_k$, equivalently they are the non-ordered lists of $n$ (non necessary distinct) elements of $\overline{k}$. Its coordinate ring is by definition $k[x_1,\ldots,x_n]^{S_n} = \{f\in k[x_1,\ldots,x_n],\forall g\in S_n,f(g(x))= f(x)\} $ the ring of polynomials that are invariant under permutation of the variables. This is the definition of the quotient of an affine variety (here $\Bbb{A}^n_k$) by a finite group of automorphism (here $S_n$).
Since $k[x_1,\ldots,x_n]^{S_n}$ is a finitely generated $k$-algebra then $\Bbb{A}^n_k/S_n$ has  a structure of affine variety. Finally with $\sigma_1,\ldots,\sigma_n$ the elementary symmetric polynomials, then $k[x_1,\ldots,x_n]^{S_n}=k[\sigma_1(x),\ldots,\sigma_n(x)]\cong k[y_1,\ldots,y_n]$ thus $\Bbb{A}^n_k/S_n\cong \Bbb{A}^n_k$, the (affine variety) isomorphism being $\bigcup_{g\in S_n} g(a) \mapsto (\sigma_1(a),\ldots,\sigma_n(a))$.
