Lucas Garron
Premium Member
So, I'm currently writing a group theory paper on enumerating the conjugacy classes of the cube group. Thought about doing it on showing that every element of the commutator group is a generator, and maybe I should have... 
Anyhow, there is a discussion here, I think. But the server is down, and I was incredibly lucky to think of saving a backup copy: http://archive.garron.us/temp/Conjugacy%20classes%20of%20the%20cube%20|%20Domain%20of%20the%20Cube%20Forum.html
So, in a simple world, we'd just calculate all the possible partitions of 8 and 12, assign each cycle a net parity in all possible ways (so that parity is satisfied), and multiply.
For 8 pieces, 3 orientations, you get {140, 130} positions for {odd, even} permutations.
For 12 pieces, 2 orientations, it's {302, 286}.
So, odd total permutations + even permutations of the entire cube: "140*302+130*286 = 79460."
...except that's wrong.
1) For edges, the totals are apparently {308, 291}. There are 11=IntegerPartitions(6) more permutations with all even-length no-net-twist cycles These only happen once each (index 0 or 1). For corners, nothing happens, by luck.
2) In addition, 10 of the even corner position and 17 of the even edge permutations are "parity-sensitive" and we need to add a category for that.
So we have {140, 130, 10, 0}.{308, 291, 17, 0} = 81120 total classes.
I haven't tried to explain that thoroughly, because it's in the linked thread. But does anyone here understand where these numbers actually come from? It seems like someone just threw in two wrenches before the final sum. After staring at the reasons for a long while, I somewhat understand why they exist, and why they happen to be the way they are. But I don't understand why exactly those two things come up, why they should happen only and exactly the way they do, and why nothing else goes wrong.
Is there some unified general understanding from which these happen to fall into confusing cases?
I'm writing a paper an this, and I don't really like the idea of saying twice "Oh, wait! Let's take a mandatory, confusing detour!" twice, with esoteric reasons for each, and no clean, intuitive motivation for them.
Anyhow, there is a discussion here, I think. But the server is down, and I was incredibly lucky to think of saving a backup copy: http://archive.garron.us/temp/Conjugacy%20classes%20of%20the%20cube%20|%20Domain%20of%20the%20Cube%20Forum.html
So, in a simple world, we'd just calculate all the possible partitions of 8 and 12, assign each cycle a net parity in all possible ways (so that parity is satisfied), and multiply.
For 8 pieces, 3 orientations, you get {140, 130} positions for {odd, even} permutations.
For 12 pieces, 2 orientations, it's {302, 286}.
So, odd total permutations + even permutations of the entire cube: "140*302+130*286 = 79460."
...except that's wrong.
1) For edges, the totals are apparently {308, 291}. There are 11=IntegerPartitions(6) more permutations with all even-length no-net-twist cycles These only happen once each (index 0 or 1). For corners, nothing happens, by luck.
2) In addition, 10 of the even corner position and 17 of the even edge permutations are "parity-sensitive" and we need to add a category for that.
So we have {140, 130, 10, 0}.{308, 291, 17, 0} = 81120 total classes.
I haven't tried to explain that thoroughly, because it's in the linked thread. But does anyone here understand where these numbers actually come from? It seems like someone just threw in two wrenches before the final sum. After staring at the reasons for a long while, I somewhat understand why they exist, and why they happen to be the way they are. But I don't understand why exactly those two things come up, why they should happen only and exactly the way they do, and why nothing else goes wrong.
Is there some unified general understanding from which these happen to fall into confusing cases?
I'm writing a paper an this, and I don't really like the idea of saying twice "Oh, wait! Let's take a mandatory, confusing detour!" twice, with esoteric reasons for each, and no clean, intuitive motivation for them.
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