Differentiable parametrizations of the permutation group

Question

Typically the permutation group is considered a discrete operation, but I wonder if there is a way to parametrize it with a larger group such that one gets intermediate continuous transformations between the discrete permutations, and such that the permutation groups are proper subgroups

Answer 1

Here's an injective homomorphism $S_n \to O(n)$: Fix an orthonormal basis $\{e_1, \ldots e_n \}$ of $\mathbb R^n$. Then for a permutation $\sigma \in S_n$, consider the linear map that sends the basis vector $e_i$ to $e_{\sigma(i)}$. Convince yourself that this is an orthogonal transformation, $T_\sigma \in O(n)$. Then convince yourself that $\sigma \mapsto T_\sigma$ is a homomorphism. So this exhibits $S_n$ as a discrete subgroup of $O(n)$.

Observe also that $\det T_\sigma = \text{sgn}(\sigma)$, so the alternating subgroup $A_n \le S_n$ is mapped into $SO(n)$.

I suppose this allows you to think of general orthogonal transformations as some sort of 'intermediate continuous transformations' between the permutations.