rational bijections on n-simplices

Question

I want to study the following bijections on the n-simplex $\{(t_1, t_2, ..., t_n) \;|\; 0 < t_1 < ... < t_n < 1\}$ $$\sigma_k(\mathbf{t}) = (t_1, t_2, ..., t_k, \frac{t_{k}}{t_{n}}, \frac{t_{k}}{t_{n-1}}, ..., \frac{t_{k}}{t_{k+1}})$$ for $k=1,2,3,...,n-1$. Each $\sigma_k$ is an involution and they generate a subgroup of the rational bijections on the n-simplex. Is there a closed-form formula or characterization of any element in this group? If $\sigma_k$ is just the permutation mapping $(1,2,...,n)$ to $(1,2,...,k,n,n-1,...,k+1)$, then I can show by induction that they generate all the permutations that fix the first entry. But for these general $\sigma_k$'s, I have no idea so far.
By the way, have people studied rational bijections on the n-simplex already? If so, I would appreciate any references!