In my algebraic geometry class my professor mentioned that $\mathbb{A}_K^n/G_n=\mathbb{A}_K^n$, where $G_n$ is the group of permutation of degree $n$ acting on coordinates.
This is not clear for me because I do not know the structure group of $\mathbb{A}_K^n/G_n $...
Can someone give me a proof of this or tell me where there is a proof that I can detail?... the quotients by group action in algebraic geometry cost me a lot.
If $G$ is a finite group acting on an affine variety $X$ with coordinate ring $A$, then the quotient $X/G$ is the affine variety whose coordinate ring is $A^G$, the invariant subalgebra of $A$. This can be your definition, or a theorem, depending on your definition of the quotient...
If $G=S_n$ and it acts by permutations on $\mathbb{A}^n$, then the coordinate ring is $k[x_1,\dots,x_n]$ and the action is by permutation of variables. Invariant subalgebra is $k[x_1,\dots,x_n]^{S_n}$: we know this as the algebra of symmetric polynomials and a basic theorem tells you that it is a polynomial algebra generated by the elementary symmetric functions $s_1,\dots,s_n$ in the variables $x_1,\dots,x$. In particular, the affine variety that corresponds to $k[x_1,\dots,x_n]^{S_n}$ is an affine space of dimension $n$.