Dynamics on Graphs 15

We give an overview of some of the remarkable aspects of the E6 story in modern mathematics (and physics!), which starts with the exceptional Lie algebra of that name discovered by W. Killing at the end of the 19th century. But in the 20th century it was discovered that this particular symmetry group had manifestations in a number of different directions, including root systems, Coxeter groups, Dynkin diagrams and lattices, but also some classical geometry having to do with the 27 lines on a general cubic surface and modern algebra with Jordan algebras. There are also important connections with representation theory.

We are taking the point of view that the underlying graph is at the heart of the matter, and that hence we should try to generate as much of this adjacent structure "internally" in terms of constructions starting with the graph. And our view is that the Mutation and Numbers Games are the key ingredient for this program. So how exactly does the number 27 appear naturally from purely graph theoretical considerations?

The answer lies with some remarkable posets associated to the Mutation Game on E6, and the concept of ideals of such a poset.

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Hello everyone I'm Norman Wildberger thanks for joining me on this adventure we're looking at E6

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the first perhaps exceptional Lie algebra slash Lie group uh associated to interesting Dynkin

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diagram it has many combinatorial aspects and actual connections to geometry and physics so

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there's a real a large story here and my general aim that I'm trying to generally convince you of

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is that we should start thinking about the graphs this graph here for example as really

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the the core of this entire story and what we want to do is we want to be able to just work

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starting with this graph generate structure that will explain much or perhaps as much as we can of

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all these interesting topics that are around this area okay so this is first of all an exceptional

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um an exceptional simple lead group or Lee algebra so these were classified by killing

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in the end of the 19th century and uh also work of carton um sort of discovered that there were four

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ongoing families of simple Lee algebras site let's say and they're also these five exceptional ones

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and the five exceptional ones are E6 E7 E8 and F4 and G2 now in our discussion so far we've talked

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about E6, E7, E8 but not really of F4 and G2 so it turns out that um that you need to go beyond

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the world of simple graphs you need to look at graphs which are not simple in other words

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where there are edges that are directed okay like roads going one way but not the other so um that's

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a little bit more complicated and we go sort of beyond the Ade graphs but in any case these five

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exceptional the algebras and the associated lead groups are fascinating objects they're certainly

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amongst the most interesting exceptional objects in mathematics and as Lie algebras

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or early groups they have respective Dimensions 78 133 87 248 for E8 52 for F4 and 14 for G2

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so we're interested in E6 which is in some sense perhaps the most accessible one of these you

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might think G2 might be but G2 is a little bit more complicated because it's associated with

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this non-simple graph so E6 generates all kinds of structure and one thing that we've already

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sort of talked about is the associated root system the positive Roots contained

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in the entire root system and that's ending up being 72 vectors in some six-dimensional space

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the associated simple Lee algebra as I pointed out here is a dimension 78 as is necessarily

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the associated compact Lee group sort of a continuous kind of group so these are

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rather onerous objects to really get your head around okay not easy at all to really understand

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um especially the the lead group aspect of this thing okay there's also an interesting lattice

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associated with this lattice in the sense of uh sort of a periodic array of vectors in in

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a vector space that's closed under addition and integer multiplication so there's a lattice and

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six dimensional space which just generated by the roots or the root populations so there's

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all kinds of interesting structures Associated to this now when you study the the Lee algebra

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um sort of abstractly one of the first things that one wants to do is to try to represent this rather

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abstract thing I have told you in a Lee algebra is but in any case to try to represent it using

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matrices because matrices are very down to earth and concrete and they connect with all this linear

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algebra that we're familiar with so it turns out that this 78 dimensional Lee algebra can be

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represented sort of in a nice way by matrices but you need to go to matrices that are of Dimension

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27 so 27 by 27 matrices okay so in some sense the sort of simplest way of explicitly concretely

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realizing these uh this particularly algebra is to write down 78 different matrices each of them

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uh a 27 by 27 matrix so it's quite onerous; in fact there's actually two different ways in some

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sense of doing this so in any case the the number 27 uh sort of figures and is intimately connected

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with E6 let me show you some other aspects of 27 that are also not far from E6 so there's

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something called an exceptional Jordan algebra okay which is also of Dimension 27. Another

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incarnation of 27 going back to classical 19th century geometry is the realization

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that a general cubic surface sort of three dimensions given by a cubic

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polynomial in x y and z such a surface generically has 27 lines lying on it the line like actually

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lying right on the surface okay now you may have to go to a complexification etc to do that but

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that's the general idea that there's somehow this number 27 is appearing in a number of ways

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so it's intriguing to think well you know it is some connection between these various 27s yeah

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that's really interesting so let's um have a look at the E6 from our point of view right which is

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to dispense with the Lie Theory or at least put it aside because it's very complicated and just

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play the mutation and the numbers games on this this graph okay and I remind you here's the graph

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okay with vertex numbering one two three four five and zero up at the top okay so in our last video

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uh we found a mutation poset where we started from a Singleton population at the vertex one and this

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was the uh the poet of the possible mutations that we could do to this initial population to increase

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the population up to the maximum population okay and we call this a mutation pole set

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and I also suggested to you that you might like to think about the associated poset of positive

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Roots okay that can actually be obtained by performing various orderings of the above

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mutations so let me show you what this thing looks like first of all its names Rplus(E6,1)

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okay so the positive roots formed by E6 where we're starting with a

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Singleton at one and the edges are labeled by mutations and this is what it looks like okay

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okay you might not be able to see all of it let's see here's the top part of it so this thing here

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is a representation of the the root population that we're starting with the Singleton with a one

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at vertex one okay and then this edge corresponds to the mutation S2 and that takes that population

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to this population so all of these things are our populations there are in fact roots in

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fact they're positive Roots they're all obtained from this one by increasing the total population

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and a certain point for example down here you can see that this diversity of options manifests

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itself we could perform a mutation as zero at this stage or mutation S4 okay so there's kind of lots

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of ways of sort of going down this path until we get to the very end which is this thing here

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okay and we could call this thing uh perhaps a positive root layer Post-it because it's not all

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positive Roots but it's sort of the positive roots that that are sliced that have the property that

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that we we're sort of starting with a one at this vertex one so all of these things in fact

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have have the property that the the one entry is one now this is a this is a lovely thing

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um so if you just observe for a second I hope you can perhaps appreciate that there's some symmetry

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to this thing right have a look at this thing okay it's symmetrical so from the top here to the uh to

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the bottom here okay it's like a string of one two three four five up here and also a string of

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one two three four five here okay sort of forms roughly a zed kind of pattern in a way Okay so

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um please uh you know investigate carefully the the relationship between this original

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mutation posted which just records the mutations themselves and this positive root layer pulls it

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Okay that's good um now I uh okay here's here's a simplified version of of the post set but you

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just see the post-set structure and you see the Symmetry a little bit more clearly just

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just post it without the labels okay so now I want to talk about ideals of a post set

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okay so an ideal of a PO said posted as a partial ordered set um suppose it has a relation given

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by this less than or equal to so it's the name of the relation that defines this partial ordered set

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then an ideal of p is by definition a subset say I obtained in P so it's a subset of the the nodes if

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you like with the property that if x is in i and y is less than or equal to X then Y is also in

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I okay so if any node is in this ideal then any sort of prior or previous node also has

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to be in the the set that's an ideal so if we look at our example here our our post set okay

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um here in red this thing here isn't ideal okay remember the the increase uh we're sort

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of reading this thing going down so as we go down we're increasing in the the partial order okay so

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this red subset has the property that if you take any one of them say that one there and you look at

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something that's less than or equal to that one say that one there then it's also in the percent

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if this one is so this is an ideal this I in red is when they deal with six elements here also in

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green is another ideal okay it's another subset of the post set which again you can check has

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the property that for any one of these guys in the set anything less than or equal to it is also in

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the set so that's another ideal has 11 elements note also that the empty set is technically also

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an ideal okay because it's sort of you know by default uh you know satisfies this this property

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okay so now we have a very interesting exercise that you will have lots of fun doing okay so I

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want you to show that this Post-it here this one right here okay actually has exactly 27 ideals

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it has exactly 27 ideas so we're meeting this number 27 purely in a completely

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combinatorial fashion just by looking at ideals in this positive root layer Post-it

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okay so it's a kind of a counting you have to construct ideals and show that there's altogether

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27 of them it's actually it's not going to take you that long but it's an interesting exercise and

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now a little bit more open-ended exercises okay so we have these 27 ideals okay so we you know so

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in principle we could you know draw 27 pictures that includes these two as particular examples

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these 27 ideals what kind of structure does this set of 27 ideals have

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okay so in mathematics when we're exploring things and we we've just discovered some some

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collection of things okay so yes we could say all right so now we have a set we've identified a set

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of 27 interesting things okay but a more subtle question is what kind of structure does this set

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have is there for example a partial ordering on them or maybe some other kind of structure

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so that's a great exercise to think about so uh I hope you're excited by the fact that that we

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somehow managed to connect with this remarkable 27 uh that somehow miraculously appears in the

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complicated uh aspects of the theory of E6 in a very simple combinatorial way just by playing

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the mutation game and looking at this positive root layer poset and the ideals contained in it