Is there a notion of dimension of representation of symmetric group $\mathcal S_n$ over $\Bbb F_2^k$? What is this dimension $k$?
In general is there a notion of dimension of representation of symmetric group $\mathcal S_n$ over $\Bbb K^k$ where $\Bbb K$ is an arbitrary field? Is there a general formula?
Are there multiple notions or is there standard notion?
I am studying this indirectly. I take a complete graph and take its Aut group which is $\mathcal S_n$. I want to label edges by vectors in $\Bbb F_2^k$ so that sum of labels over simple cycles are not part of kernel. I want to know the minimum $k$ that can do this. I am thinking dimension of aut group is same as $k$. I am trying to show this.