Dynamics on Graphs 24

28 views May 5, 2023

We use a fundamental combinatorial construction to start with a very simple poset with just 3 elements, and then generate much of the ADE hierarchy in a remarkable fashion. This completely avoids traditional root system or Lie algebraic ideas, and also finesses the Mutation and Numbers games, although as we shall see that construction ends up having intimate connections with all of these structures. Many interesting lines of investigation are suggested by this exploration: again ideal stomping ground for amateur mathematicians and indeed computer programmers. Here is a link to the pdf of this lecture: https://www.dropbox.com/s/12nz6y65nfw...

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Hello everyone, I’m Norman Wildberger. Thanks for joining me on this adventure. Today,

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we’re going to look at the ADE graphs in a quite novel combinatorial fashion where we inductively

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approach their story starting from a very simple poset and just performing this operation that we

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looked at in our last video, that of taking ideals of a given poset. So, we start with a poset and we

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consider the lattice of ideals of that poset and that’s naturally also a combinatorial structure

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on lattice which is a kind of poset. And so then we can iterate that and take J of that

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and if we start with this particular poset here called P, just a three-element poset with a little

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two-chain, A, B, and another element C, then we’re going to see that the ADE story sort of emerges

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simply by taking recursively inductively ideals starting with this poset P. So let’s see how that

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works and I want to emphasize that what we’re doing here is independent of the mutation game,

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at least in principle, although we’ll see that it strongly connects with what we’ve learned about

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the mutation game and the Dual mutation games. And in particular, what we’re talking about

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today will overlap significantly with lectures 14 and 15 in this series on the E6 graph and the

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connection with the number 27. And at the end of that last video, the 15th, I gave you some

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exercises to compute some things and you’ll see that I’m basically going to answer that

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question here but again completely independently of the mutation game story. Okay, so let’s start

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taking ideals of this thing. Okay, here they are right here at the top is the empty set okay which

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is always an ideal and then we have the set poset generated by A generated by just C and then A and

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C together so that would be these two that subset and over here B that would be this subset here and

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finally at the bottom the ideal generated by both B and C which is all of everything here okay so we

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get these six ideals forming this rectangular poset which is a product of chains there’s a

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three-chain this way and a two-chain this way so this thing has six elements and to go to the next

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step it’s convenient for us to rename the elements here so let’s just rather arbitrarily rename them

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as X Y Z U V and W. And so now we’re going to take this as a starting poset and take J of it.

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Alright well what do we get? So here is what you can see and again this thing here

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so again empty set at the top the ideal generated by X and then below it the ideal generated by Y

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and also the ideal generated by U below that well the ideal generated by Z that’s that one there

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or the ideal generated by Y and U which is that triple there and then Z U is the ideal consisting

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of these four points while just ideal generator by V is this little quadrilateral here and then

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Z V is these Five Points here at the bottom W is the entire thing so we get this Pleasant kind of

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uh like a dart kind of shape which has 10 elements and again because we want to keep on going further

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we’re going to rename the entries in this fashion E F G H I J K L M and N

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okay so what happens at the next stage well we have to take ideas of this so here is the post

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set or the lattice that we’re starting with now this is J squared of P and then J cubed of P is

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going to be this lattice here so have a look so there’s um the empties at the top and to make

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things a little bit more streamlined let’s leave out brackets and commas from now on alright so

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this G just represents the ideal generated by G over here so that would be that triple while G I

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represents the ideal generated by G and I so that would be those four points there Etc so you can

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check that here are the ideals of this thing here right so for example H J the ideal generated by H

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and J that would be these six elements here while KL B these I guess eight elements here

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Etc now this is a very pleasant little um thing Um, and someone arbitrarily let me call it the

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bat, okay because maybe it has a vague bat-like resemblance. And uh, so that’s J cubed of P and

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it has 16 elements. Next on our list, well to keep going we’re going to first rename the elements

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here so let’s call them A B C D E F G H I J K L M N O P. Okay, now the ideals of this thing,

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well so there it is there. So we’re starting with the empty set again, the ideals generated by A B

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C D going down that first diagonal there okay and then we have these various other ones and

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it’s really a great exercise to sort of stare at this and to first of all admire it. Okay,

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I call this ideal the swallow and it has 27 elements and this is the answer to an exercise

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that I gave you in that earlier video where I asked you to compute the ideals of this thing

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and show that there were 27 of them and actually to show what the structure of those ideals are.

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So here are those 27 elements but now as a partially ordered set and let me remind

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you that that 27 connects with various other interesting things in mathematics, in particular,

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the 27 lines on a cubic although we haven’t talked about that yet. Okay so let’s call this thing the

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swallow okay this is a beautiful thing it actually arises in physics too in conformal field theory

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um so it’s a really remarkable partially ordered set but we want to go one further okay so

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um let’s rename the swallow but maybe before we do that we’ll make an interesting observation that

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the top layer of these lattices are so far A4 D5 E6 and E7 so let me we’ll see what I mean by that

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so up here uh in our first J of P okay if we just look sort of the top part like if you’re looking

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down on it okay then you uh you see this Z Y X uh shape that’s an A4 graph if we go to J squared

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of P and look down on it well we look down on it but including this I here so that top part there

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okay that’s the D5 graph and then with the bat top part here including that F that was E6 okay so we

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seeing the ad graphs appearing here and over here we can see E7 appearing right up at the top here.

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So that’s one connection but of course, the connection between these posets and the ADE

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graphs is much deeper than that you know these are very deep connections uh that we’re just touching

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with at this stage okay so to go one step further let’s rename the swallow so here is renaming

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um it’s got 27 elements there’s only 26 letters in our alphabet so let’s uh also have zero here

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so there’s zero then A B C D etc down to Z so we’ve renamed the swallow or J

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to the fourth of P okay and now well here’s J to the fifth of P okay.

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Okay, so here is what we get when we take the ideals of the swallow,

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this thing here take the ideals of that.

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We get this thing, okay let me make it just a little bit smaller so you can see it all,

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it’s almost all going down to Y Z okay and I’ve used two different colors so you sort of see a

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black layer here and then underneath it the sort of symmetrical green layer and they’re sort of

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connected with one of these cubes sort of seen this kind of thing before in the previous video uh

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where some there’s some vertical bonds so it has a somewhat of a three-dimensional aspect and we also

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know that the top layer here is E8 so we’ve got E8 here and how many elements does this thing have?

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Well, it has 56 elements and as will turn out I will see that 56 actually is a pretty important

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number in the ADE story just like 27 is in fact and this is a remarkable thing in this eagle. And

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I claim that you know a lot more sophisticated um aspects of the ad story especially having to do

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with representation theory and the associated Lie algebras are actually embodied in some remarkable

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perhaps even miraculous way uh with these kind of posets, these posets which are actually lattices

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because they’re J's of other posets. In fact, they’re very strongly lattices because they’re

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J's of J's of J's of posets etc. And this one here is uh just a beautiful thing and of course there’s

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a natural question you know what’s the next one in this sequence so we can see this massive research

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program ahead of us because we’ve just started with this very simple poset and we’ve created

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all this structure but we’ve only just gone you know up to the E8 level but in principle,

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we could keep on taking J's of this thing and getting bigger and bigger and bigger structures

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that are in some sense you know built up from really simple foundations.

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So it’s not so dissimilar say from the kind of investigation say that Stephen Wolfram

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was interested in with cellular automata starting with a cellular automata with very

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simple rules and then running this thing and generating you know massive complexity

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at least in some cases. So here’s something similar is happening but it’s it’s not really

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you know an algorithmic thing so much it’s just that we’re we’re having a ladder okay

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and we can go up this ladder and as we go up the ladder things just get more and more complicated

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and you know there’s all kinds of questions and things to ask. These are very interesting

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lattices and so we get all kinds of questions that we can and we will ask about these things but in

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particular the sort of obvious one is which one is next? What happens if you take J of this thing?

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What does it look like? How many elements does it have? So that’s now starting to get to be a

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rather serious challenge and I think it’s one that you will have fun doing. You can do this

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by hand but also maybe even some programmers might think about how to you know how to do

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this sort of more systematically so that we sort of input a poset and then start creating

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J's of those posets or ladders of those J's. So in principle, this is you know

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a very programmable thing although you do end up having to sort of uh you know picture what’s going

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on in this sort of two-dimensional fashion perhaps but maybe there are other ways of representing

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outputs as well. So we’re touching base with really interesting stories and as we’ll see this

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has huge connections with more sophisticated things but we’re already seeing remarkable

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structures uh happening and really starting with very simple combinatorial objects really

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just sort of the elementary graphs or posets and playing relatively simple games with them.

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Thank you.