The way Lachy talked about \(S_n\) made it seem like there is a canonical way to think of the permutation group \(S_n\) as a scheme. Perhaps this is one way:
if \(H \subset G\) is a normal subgroup of \(G\), and \(G\) is an algebraic group, then I can define the polynomial functions \(\mathbb C[G/N]\) on \(G/N\) by \[\mathbb C[G/N] := \mathbb C[G]^N.\] Then perhaps you could define \(\mathbb C[N]\) by \[0 \to \mathbb C[G]^N \to \mathbb C[G] \to \mathbb C[N] \to 0.\]