ScalingStacks

The permutations of ${\mathbf{Z}}$ can be described as collections of strands in $[-1,1]\times{\mathbf{R}}$ going leftwards from integer points on the vertical line $x=1$ to integer points on the vertical line $x=-1$. Thanks to their $n$-periodicity, those permutations that are elements of $\hat{{\mathfrak{S}}}_{n}$ can also be encoded in a collection of strands drawn on a cylinder, going from right to left, by passing to the quotient of the vertical strip $[-1,1]\times{\mathbf{R}}$ by the vertical action by translation of $n{\mathbf{Z}}$.

Here are some elements of $\hat{{\mathfrak{S}}}_{3}$:

The multiplication $\sigma\sigma^{\prime}$ of $\sigma$ and $\sigma^{\prime}$ in $\hat{{\mathfrak{S}}}_{n}$ corresponds to the concatenation of the diagram of $\sigma$ put to the left of the diagram of $\sigma^{\prime}$ as in the following example:

The defining relations for $\hat{{\mathfrak{S}}}_{n}$ are depicted as follows

The elements of ${\mathfrak{S}}_{n}$ correspond to diagrams whose strands do not go in the back of the cylinder, hence can be drawn on a rectangle. For example, $s_{12}$ above can be represented as follows: