Is there an error in the formula of S4?

[bblfish]

As far as I can tell the above picture tells me that (0 1)(1 3 2)=(0 1 3 2) but from my calculations it is (0 3 2 1) which matches the transformations. See my calculations below. It also matches the transformations at the nodes. I start with an sequence 0 1 2 3 and calculated one whole triangle at the bottom, and it seemed to work. The permutations in the top triangle work if I use (0 3 2 1), and in the bottom right corner the square calculation seems to agree with it.

[nathancarter]

Here's the calculation that GE is doing:

First, note that (1 3 2) means the permutation mapping 0->0, 1->3, 2->1, and 3->2.

Second, note that (0 1)(1 3 2) is a permutation that acts similar to how composition of functions acts (since permutations are functions), in that it expects its argument to be on the right, and applies the right-hand permutation first, then the left-hand one.

Putting those together, we have this:

• What does (0 1)(1 3 2) do to 0? First, (1 3 2) maps 0->0, then (0 1) maps 0->1. Answer: 0->1.
• What does (0 1)(1 3 2) do to 1? First, (1 3 2) maps 1->3, then (0 1) maps 3->3. Answer: 1->3.
• What does (0 1)(1 3 2) do to 2? First, (1 3 2) maps 2->1, then (0 1) maps 1->0. Answer: 2->0.
• What does (0 1)(1 3 2) do to 3? First, (1 3 2) maps 3->2, then (0 1) maps 2->2. Answer: 3->2.

So the answer is 0->1, 1->3, 2->0, 3->2, which we write as (0 1 3 2).

I think GE is correct. Does this help?

[kcrisman]

For what it's worth, I agree with @nathancarter - but perhaps it should be made more clear in a clickable link or something on the visualization that you are using the right-to-left convention? Maybe a "permanent footnote" for permutation groups. I know my students asked about which direction to multiply permutations the very first day we introduced them.

[nathancarter]

Good point; we should have this in the help somewhere.