Dynamics on Graphs 14

Let's illustrate the Mutation and Numbers Games on the ADE graph known as E6, which is in some sense the simplest of the five exceptional ADE graphs, and is connected to a rich variety of interesting mathematics and indeed physics. The main aim here is to construct the mutation poset arising from starting with a singleton population at the end of one of the two long tails of the graph.

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0:00

Hello everyone I'm Norman Wildberger thanks for joining me on this adventure we're looking at the

0:05

mutation game the numbers game on graphs thinking about creating a theory of exceptional structures

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in mathematics and physics starting from the graphs with these two relatively simple games

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which I'm claiming generate so much structure and I'm in the process of generating some of that

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initial structure today we're going to look at a very interesting sort of First Look at E6 it's

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one of the five exceptional simple Lee algebras or simple groups are found by killing at the end of

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the 19th century and has gone on to have all kinds of other manifestations that I'll talk about more

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in a future video today what we want to do is we just want to make an initial foray into this

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into this remarkable World by actually playing the mutation and the numbers games simultaneously okay

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so that you can see the important correspondence and the correlation between these two games

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okay so our labeling is this one here familiar I hope uh there's the E6 graph one two three

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four five are the vertices in the base and then this one up here is zero and when we

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use solid dots or nodes that's typically when we're describing populations and when we use

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open nodes uh we'll be talking about weights which is sort of on the Dual side on the on

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the uh numbers game side okay so we're going to start as we usually do when we're generating roots

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with a Singleton population and we're going to choose to start with a Singleton population

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at this vertex one right here so there it is there there's the starting point

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okay population of one at the vertex one now the first thing to observe is that the corresponding

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weight okay over here is given by this with a minus two here A 1 here and zeros everywhere

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else and let me remind you that these numbers here correspond to what we can expect to happen

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to this population if we mutate at the various vertices so if we mutate at vertex one then the

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one changes to minus one so that's a change of minus two if we mutate that vertex 2 then that

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zero becomes a one okay so a change of plus one and mutations elsewhere don't move anything okay

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so that's our starting point and when we look over here on the weight diagram side we can see where

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we can mutate in order to increase the population that's going to correspond to entries which are

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strictly positive in this case there's only one of them this one here this one here that's telling us

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if we want to increase the population of this thing here by a mutation we have to mutate at

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the vertex 2. so that's what we're going to do we're going to perform the mutation S2

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okay so what happens then well on the mutation side well of course that zero just changes into

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a one but correspondingly over here what happens well you get this weight minus 1 minus 1 1. and

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purely in terms of the weights over here what's happened is that this one okay has propagated to

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the its direct neighbor so we've added one to the both neighbors of this vertex two and then

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the value has negated so the one has gone to all the neighbors and then the one has changed

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into a minus one okay so now we're in a position to see that the Next Mutation had better be at

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vertex three because that's the only strictly positive value that we're seeing over here on

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the white side so let's agree to mutate at vertex three so we're going to perform S3 what do we get

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well that zero changes into a one and correspondingly on the numbers game side

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okay that one there propagates to all its three adjacent neighbors namely to this vertex to this

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vertex to this vertex so we get a zero there a one there a one there and then finally the one

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that was there originally changes sine to a minus one okay at this stage here we have two possible

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places where we can mutate to get an increase because there's two strictly positive entries one

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at vertex zero one at vertex four so here's the place where the the road kind of diverges we could

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either do this one or we could do this one and so let's agree to do S4 rather arbitrarily okay

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and when we do S4 well then we're going to pick up another one over here and the corresponding weight

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is going to change as follows the one is going to propagate to its neighbor so that minus 1 is going

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to become a zero that zero is going to become a one and then it changes sign 1 goes to minus one

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now again there's still two places that we can further mutate to increase the population namely

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of Zero Hour at five that's to S5 so we do S5 then this zero changes into a one and correspondingly

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this one is added to the minus one to cancel that out and the one changes into a minus one

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at this stage here there is now no more choice

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as to a mutation that's going to increase we have to mutate at zero

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okay I'm going through this so rather carefully in detail I really want you to follow the argument

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okay it's really important that you follow and understand completely this correspondence

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between what's happening on the mutation side what's happening on the numbers game side

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Okay so we've just mutated at um at zero so the uh the one well over here what happens is that we

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get a one over here and then over on this numbers game side we the one moves over there so we get a

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one there and changes sign where it started minus one there at this stage here there's also only one

5:54

more option we have to mutate at vertex three so we apply S3 then over here uh what's going

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to happen is that one is going to change into a two because on the mutation side what happens is

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that you negate the value and add the sum of all the neighbors so the one becomes a minus one and

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we have to add three for a total of two so yes we really have increased the population okay and

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um and then on the dual mutation side what's happened when we've done S3 is that this one

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has propagated okay to there and to there and cancel with the minus one that was up at zero

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to give us zero there and then the one changes to a minus one at this stage then there's two

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possibilities for increase namely vertex two or at vertex four so that's rather arbitrarily

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choose vertex two okay so we're going to mutate to two so that one is going to become a two

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and this one over here is going to propagate to its neighbors canceling them out

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the one turns into minus one there's a bit of a cut here don't worry about that that's still a

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zero at the vertex zero up here at this stage here there's only one choice S4 that will increase the

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population so let's mutate S4 that means that that one is going to turn into a two that one

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over here is going to add to its neighbors a one there a zero there and change signed per minus one

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at this stage we can only mutate at vertex three so let's do that then this 2 changes into a three

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because minus two plus five is three and over here the one changes into a minus one but before it

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does that it adds itself to all its neighbors so we get zero here zero here and one here

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and now there's only one place where we can mutate to increase the population namely at

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vertex zero so we mutate there that means that one changes into a two and this one is

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added to the minus one to give us zero and change a sign to give us minus one

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and at this stage we see that the process terminates because there are no more strictly

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positive values over here on the weight diagram side okay no strictly positive values that means

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that no mutation is going to increase the population this is the maximum population

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and I remind you that that's a rather special phenomena that happens for the Ade graphs

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right that's typical of the Ade graph story that these mutation gain these trying to increase the

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population it stops with a maximum population it's not what happens for a typical graph okay but here

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the whole thing is stopped and so then we get a sort of a finite structure so we now we can

8:43

sort of review what we've done we can go back up here okay and just record all the mutations that

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we did starting with S2 and then S3 and then S4 and then S5 and then s0 S3 S2 S4 S3 and s0 so

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we record all that let's say the sequence here s two three four five zero three two four three zero

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okay so that was the sequence of mutations that we did starting with a Singleton at one at the

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vertex one and now we know that we can replace this sort of a somewhat arbitrary linear order

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that's appearing here with a more or a richer a more honest representation of what's going

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on that's that takes into consideration the choices that we've been able to make and that's

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a tank here by looking at this this um mutation pulse set okay so what is that

9:43

well we just you know label these things so there's two maybe I should Circle this one okay

9:50

that's sort of the starting point and then we perform mutation two and three and then four

9:54

then five then zero then three then two then four then three then zero but we sort of um do it this

10:00

way because if you if you go through it and you know at this stage here when we had mutated at

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three we had a choice of mutating either at zero or a four so we happen to do four first but you

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can also have done zero first but crucially you can't mutate a second time at vertex three until

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you've done both the zero and the four mutation at this level so this post set is recording the

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the Necessities that are required okay in order to make this mutation you have to have done all the

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the previous ones in other words all the ones that are earlier or or higher up in in this post set

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it's not a linear order it's a partially ordered set structure

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now uh so that's great we'll stop here there's going to be a lot more to say about this and

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we're going to continue with our in our next video but um you know when we were looking at D5 what we

10:55

did then is we looked at a corresponding post-set of positive Roots okay the corresponding posted of

11:01

positive Roots we can call R plus e61 so in other words we look at all the the root populations that

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we can obtain from the Singleton of the Singleton that one by performing these increases in other

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words sort of by going through these through this post set sort of in all possible different ways

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and you might very well like to try to create that post set that's a really excellent exercise okay

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um to create this Post-it and see what it looks like and admire it.