Use the idea of stacking diagrams to show that $S_n$ is generated by $\{(i \text{ } i+1):1\leq i \leq n-1\}$.
In the book Office Hours with a Geometric Group Theorist, the authors give the following example of a "stacking diagram" (basically the same as a braid diagram):
I feel like I intuitively understand that stacking such diagrams can yield any permutation of $S_n$, but I'm not sure how to rigorously prove that $S_n$ is generated by transpositions $(i \text{ } i+1)$ using this concept explicitly.
Rather informally, I think it could go something like: Since any $\sigma \in S_n$ is the product of $2-$cycles, consider any $2$-cycle $(n \text{ } m)$ (with $m>n$) in the decomposition of $\sigma$ into transpositions. Then the stacking diagram corresponding to $(n \text{ } m)$ can be expanded such that each crossing in the diagram changes $n$ to $n+1$, then $n+1$ to $n+2$, and so on, with the final crossing being $m-1$ to $m$. Then take this compound stacking diagram, and stack it along with the other such diagrams for the remaining transpositions in the decomposition of $\sigma$, and the resulting diagram will identify $\sigma$ as a product of elements $(i \text{ } i+1)$ for $1\leq i \leq n-1$.
How can I fill in the details here to make this proof complete?
Best Answer
Let $x_1<x_2<\cdots <x_n$ and $y_1<\cdots <y_n$ be real numbers. Then, for a permutation $\sigma\in S_n$ your diagram can be realized by drawing a line segment $\ell_i$ between $(x_i,1)$ and $(y_{\sigma(i)},0)$. Now, for each $1\le i<j\le n$ let $(a_{ij},b_{ij})$ be the intersection of the line segments $\ell_i$ and $\ell_j$ (if it exists). If $\{x_i\}$ and $\{y_j\}$ are chosen appropriately, the $b_{ij}$ can be taken to be all distinct (so, in particular, there are no multiplicity-three intersections).
Order the $b_{ij}$'s as $b_{i_1j_1}>\cdots>b_{i_mj_m}$. Then, inductively define $\tau_1,\dots,\tau_m$ by $\tau_r:=(\sigma_{r-1}(i_r)\sigma_{r-1}(j_r))$, where $\sigma_r:=\tau_r\cdots\tau_1$. Then, one can prove $\sigma_{r-1}(i_r)$ and $\sigma_{r-1}(j_r)$ always differs by $1$.