When trying to understand something I identified as a some kind of permutation group, I came up with the following visualization as a graph, using the fact that both the current "configuration" of $n$ distinct elements as well as the "group action" / permutation itself can be represented by the same thing (an ordered list of $n$ numbers), so:
- start with the "initial configuration" $[1, ..., n]$ as the node
- for each node, apply each available permutation, adding the resulting list as a node and also labelling the arrow by the permutation, e.g. $[0, 1, 2]\overset{[1,2,0]}{\rightarrow} [1,2,0] \overset{[1,2,0]}{\rightarrow} [2,0,1]$
- repeat until no new nodes or eges are added.
I just know the very basics of category theory, but to me it looks like I defined the category induced by the group (or something like that), with the peculiar property that I can "identify" a node with a set of edges in the same graph.
Next thing I noticed was how e.g. the trivial configuration $[1..n]$ corresponds to the identity arrow of each configuration, and similarly, one can find "cycles" following the arrows labeled by the same permutation gives some "orbits" which in the case of starting with the "identity" configuration yields the "cyclic subgroup generated by the group element", etc.
I played around with this and noticed that apparently(?) you can construct all groups $S_i$ using a rather simple construction:
$i=1:$ trivial group $C_1$ consisting of the identity element and identity arrow, that can be represented both by the list $[1]$
$i\to i+1:$
- take $i+1$ copies of the previous level (so for each node $n$ from $C_i$, now there are copies $n_1 ... n_{i+1}$)
- add a pair of inverse actions $rotl_{i+1}$ and $rotr_{i+1}$, and cyclically connect all copies of the same node with them (in case of $i=2$, both are the same and correspond to $[1,0]$, i.e. $swap = rotl_1 = rotr_1$), i.e the directed edge cycles $n_1 \overset{rotl_{i+1}}{\rightarrow} n_2 \overset{rotl_{i+1}}{\rightarrow} ...$ and $n_{i+1} \overset{rotr_{i+1}}{\rightarrow} n_i \overset{rotr_{i+1}}{\rightarrow} ...$ for each distinct $n$ from the previous level
So the whole construction basically adds another "orthogonal cyclic degree of freedom" to switch between the copies of the previous level. I have not figured out a fully formal definition on level of the lists I use as labels (the "permutations"/"configurations"), but from the graph view its just cyclically connecting copies of the lower levels, and each level has $n!$ nodes, so it looks like a kind of recursive construction to yield the symmetry groups.
I found this quite interesting and now I am curious about both of these things:
- Is there a proper name for this peculiar category-like labeled graph representing the group?
- Is what I am constructing actually the symmetric groups $S_n$, or something else? Does the construction also have some name? To what "generator base" do the edge cycles/action that I add in each level correspond?
I've no background in group theory or abstract algebra beyond the basics and actually stumbled on this playing around with some operations while doing something totally unrelated to group theory, but now I'm quite curious about what I have "found".
So some pointers to relevant (I guess basic) literature or a "proper" explanation of what I was "actually" trying to describe would be great, because it feels like this must be something very simple and well-understood. I'm sorry if this is still way too unclear and informal, but that's the reason I am asking for help clearing it up... Thanks in advance!
EDIT
I'll try to clarify the construction going from $S_2$ to $S_3$ (is there some (limited) TikZ support here? that would make it much easier)
On top, we have the action groupoid for $S_2$, unlabeled and labeled.
On the bottom - there are 3 copies of the $S_2$ graph, but we change the interpretation of the labels - instead of using $[0,1]$, we use $[0,1,2]$, and make the "original arrows" act only on the first two elements. This "swap" action I labeled $l_2$ for left rotation of prefix of length 2, while the new edges are the left rotation on prefix of length 3 (i.e, the full list). Technically, the identity arrows could also be called $l_1$ for consistency.
So the way I see it, the $l_3$ operation actually implicitly "picks" a different single element to "fix"/move out of the "scope" of the other operations on the lower level (in this example, $l_2$ and the identity). I left out transitive edges/inverse arrows for clarity.
Cool ideas! It sounds like you're looking for the concept of an action groupoid: https://ncatlab.org/nlab/show/action+groupoid. This is a category you get from letting a group act on a set. Specificially, I think you're describing the action groupoid of $S_3$ on itself.
You think of $S_3$ as a set and a group in your question, so it may help to distinguish the two viewpoints. Here's a set $S$ of the orderings of $\{1,2,3\}$, written in your notation: $$S = \{[0,1,2], [0,2,1], [1,0,2], [1,2,0], [2,0,1], [2,1,0]\}$$ Now let $G$ be the group $S_3$ consisting of all the bijective functions $\{0,1,2\} \to \{0,1,2\}$, with the operation of composition. For instance, a function $\sigma$ sending $0 \mapsto 1$, $1 \mapsto 2$, and $2 \mapsto 0$ is one element of $G$.
We can define an action of $G$ on $S$, which basically means interpreting elements of $G$ as permutations of $S$. For example, if $\sigma \in G$ is a permutation function, and $[a,b,c] \in S$ is an ordering we set $\sigma([a,b,c]) = [\sigma(a),\sigma(b),\sigma(c)]$. For instance, with $\sigma$ defined as above we have $\sigma([0,1,2]) = [1,2,0]$. (It's a good exercise to check that this is a valid group action.)
What you've done is use this group action to identify permutations in $G$ with orderings in $S$, matching $\sigma \in G$ with $\sigma([0,1,2]) \in S$. In this way, you can view $S$ and $G$ as being the same set, so it's $S_3$ acting on itself. Now, you can return to the nLab article on action groupoids, and see the definition matches.
I don't quite follow your construction in the next part, but I think you're basically noting that an $n$-element set has $n$ subsets of size $n-1$. If we view $S_{n-1}$ as acting on each of these independently, we get $n$ different action groupoids. For instance, in $\{0,1,2,3\}$ we could view $S_3$ as acting on the subset $\{0,1,2\}$ or $\{0,1,3\}$ or two other possibilities. When you draw your 4 copies of the diagram for $S_3$, you get one for each of these sets.
The subsets $\{0,1,2\}$ and $\{0,1,3\}$ are interchanged by the permutation $\tau$ sending $2 \mapsto 3$ and $3 \mapsto 2$. So, if you draw four copies of the action groupoid for $S_3$, corresponding to three-element subsets of $\{0,1,2,3\}$, you can next add an arrow for $\tau$ and it will connect two of the diagrams. Adding more transpositions (permutations swapping two elements) you can connect all four of the diagrams. The action groupoid for $S_4$ will be the groupoid generated (in some sense) by the objects and arrows you've drawn.