Dynamics on Graphs 12

The Mutation and Numbers Games are basically the most important games you can play on a graph, and their study leads to immeasurably rich combinatorial, geometric and algebraic structures, including much of the representation theory of compact simple Lie algebra and Lie groups and their affine variants, Coxeter groups and root systems.

Crucially these two games are dual in a precise sense, and we want to explain this duality explicitly in this lecture by working through a concrete example: the case of the ADE graph D5.

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Algebraic Calculus One (for the Open Learning Course) : https://www.youtube.com/playlist?list...

Algebraic Calculus Two (for the Open Learning Course in preparation): https://www.youtube.com/playlist?list...

Advice for Mathematics Exploration (for Members only): https://www.youtube.com/playlist?list...

Solving Polynomial Equations (Members only): https://www.youtube.com/playlist?list...

Hexagrammum Mysticum (Members only): https://www.youtube.com/playlist?list...

Algebraic Calculus and de Casteljau Bezier curves (Members only): https://www.youtube.com/playlist?list...

Exceptional Structures in Maths and Physics via Dynamics on Graphs (Members only): https://www.youtube.com/playlist?list...

Maxel Inverses and Orthogonal Polynomials (for Members only): https://www.youtube.com/playlist?list...

Triangle Centres: https://www.youtube.com/watch?v=iLBGX...

Six: An elementary course in pure mathematics: https://www.youtube.com/playlist?list...

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Maths and Music: https://www.youtube.com/playlist?list...

Diffusion Symmetry: https://www.youtube.com/playlist?list...

Elementary Mathematics (K-6) Explained: https://www.youtube.com/playlist?

list=PL8403C2F0C89B1333

Playing Go: https://www.youtube.com/playlist?list...

Year 9 Maths: https://www.youtube.com/playlist?list...

Ancient Mathematics: https://www.youtube.com/playlist?list...

Wild West Banking: https://www.youtube.com/playlist?list...

Sociology and Pure Mathematics: https://www.youtube.com/playlist?list...

Old Babylonian Mathematics (with Daniel Mansfield): https://www.youtube.com/playlist?

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Math History: https://www.youtube.com/playlist?list...

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Famous Math Problems: https://www.youtube.com/playlist?list...

Probability and Statistics: An Introduction: https://www.youtube.com/playlist?list...

Boole's Logic and Circuit Analysis: https://www.youtube.com/playlist?list...

Universal Hyperbolic Geometry: https://www.youtube.com/playlist?list...

Differential Geometry: https://www.youtube.com/playlist?list...

Algebraic Topology: https://www.youtube.com/playlist?list...

Math Seminars: https://www.youtube.com/playlist?list...

0:00

Hello everyone I'm Norman Wildberger thanks for joining me on this Dynamics on graphs adventure

0:06

and I want to clarify the distinction between the mutation and the numbers game is these two

0:12

beautiful games which are very closely connected they're sort of dual to each other that we can

0:17

play on a very general graph and these games I claimed generate so much of what's currently

0:23

considered representation theory of associated with cognitorics of root systems so I want to

0:29

explain that to get you going in this direction by looking at an example okay so we'll look at

0:34

the graph D5 and we're going to play the mutation game and we'll see how correspondingly the weights

0:41

that are you know that are generated by this Duality a change according to the rules of the

0:46

numbers game so on one side we have mutations and another we have so these dual mutations or

0:52

firings Associated to the numbers game so here's the graph D5 let's just all get on board with this

0:59

convention that our vertices are labeled one two three four with the zero up on the top here okay

1:04

and the population is going to be expressed starting with the p zero value and then P1

1:10

p2p3 B4 as a row Vector corresponding weights are going to be indicated on a diagram that

1:15

looks very much the same except that there's going to be open vertices to distinguish it

1:18

from the sort of population picture okay and the corresponding function here called a weight will

1:25

be expressed with a column Vector w0w1 Etc but we won't actually have occasion to use these um in

1:33

this lecture we're just going to work directly with the diagrams and see what happens when we

1:37

perform mutations so we're going to start with a Singleton population and then we're going to move

1:42

up to the maximal population applying as usual the various mutations corresponding to the vertices

1:49

great okay so the starting point is right here okay with this Singleton population

1:55

that has a one at this vertex remember that's vertex number one so they as a population

2:01

that would be represented by this vector and over here is the corresponding weight

2:09

vector and let me remind you that these values here correspond to the expected increments that

2:16

we could get by applying all possible mutations to this population so if we applied the S1 okay

2:28

then this one would change to a minus one so for a total change of minus two that's where there's

2:32

a minus 2 here if we applied the S2 operation corresponding this vertex here then the zero would

2:39

change to a minus zero we'd have to add one so the zero would move to a one so it would be a change

2:44

of plus one that's why there's a plus one here and mutations at the other words these are not going

2:49

to change anything that's why there's zeros there okay so this is this basic correspondence between

2:55

a mutation and this dual mutation okay now if we're interested in increasing the population

3:03

over here okay we can look to these various values here and just choose the vertex where the biggest

3:10

number occurs because that's telling us where the biggest change is going to be happening

3:15

here in fact we see there's only one place that will result in a positive change a strictly

3:20

positive change because of this value 1 at this vertex here so we know that if we want

3:26

to increase this population then we should mutate at this vertex here so it's Central

3:32

vertex which is vertex two so that's what we're going to do we're going to apply S2

3:38

and I've represented that here by these red circles telling us you know what we're going

3:43

to do next to increase the population so what happens to our population if we

3:49

mutate at S2 well the zero changes into a minus zero we have to add some of the

3:54

neighbors so we have to add one so this zero becomes a one and this is the new population

4:00

at this stage here you can check that the corresponding

4:04

weight over here is given by this thing here so now this is the updated okay any sort of

4:12

increment profile that tells us what the various mutations on this population would actually

4:20

result in okay mutating here would give you a decrease of minus one uh retaining here would

4:28

give you a decrease of minus one but mutating here or here you would get an increase of one

4:35

so if we're again interested in increasing the population we should subsequently mutate at one

4:42

of these positive values but here crucially there are two possible choices so so the the

4:48

three of possibilities Forks here okay so let's choose rather arbitrarily say this vertex here

4:57

vertex number three okay so we're going to decide to perform S3 well what happens then

5:06

mutate here okay that zero changes into the minus zero we have to add one so we get a one there

5:12

and then the corresponding um weight Vector is is this one here now importantly all right and

5:19

the whole point of the numbers game is that we could have gone from this weight Vector

5:23

to this weight vector and then to this weight Vector without reference to these

5:28

corresponding mutations we can done this just at the level of these weight vectors by playing the

5:34

the numbers game a different kind of mutation and there the rule is that this mutation S2 at this

5:41

vertex here is going to change this one into a minus one and it's going to add a copy of whatever

5:47

is here to its directly adjacent neighbors the minus 2 becomes a minus one the zero becomes a

5:53

one and the zero becomes a one and similarly going going from here uh to here with this

6:02

dual mutation at the vertex three well the one changes to a minus one and the directly adjacent

6:09

neighbors get one augmented to them so this changes into a one and this minus one becomes

6:14

a zero okay and then we can carry on again at the stage here we have also a choice we could

6:21

mutate at this vertex like this vertex let's choose this one here okay that's S4 the result

6:29

is that the population is going to change into this population here is a corresponding weight

6:34

Vector you can check okay and at this stage here now a subsequent increase is only obtained by

6:42

mutating at the vertex zero because that's the only positive value so if we're interested in

6:47

increasing which we are then at this stage our hand is forced we have to perform at s0

6:53

what's going to happen well the serum's just going to turn into a 1 okay but at the level

6:58

of the weights this one changes into a minus one and we have to add that one to the neighbor

7:04

namely this one here so the a 1 appears down here at the vertex vertex number two

7:11

okay and then at this stage again there's only one possible positive

7:16

value here so we're forced to mutate at vertex two that results in this one changing into a

7:24

two and the corresponding weight diagram changing to this one here so that's one sort of propagated

7:30

to its neighbors and and indicated itself and now subsequently here there's only a single

7:35

positive one so again our hand is forced if we want to increase things we have to do an S3 and

7:41

that S3 is going to change this one into a two and over here in the weight diagram the one is

7:46

changing into a minus one and when we add one to the directly adjacent neighbors we get all zeros

7:52

so we've kind of you know gone from one extreme to another and this in some sense we've kind

7:59

of dumped in some sort of Full Circle except We've Ended up in a situation where we've got

8:05

all zeros except for a single minus one but in particular there are no positive

8:09

values so there's no more increases possible and that's why this is the maximal root population

8:17

great so please go through that and and and check that you you really understand all these steps

8:23

okay it's really really useful and all together we've performed mutations as two and then S3 and

8:30

then S4 and then as zero and then S2 and then S3 and so we can represent that by this list reading

8:36

left to right S2 S3 S4 s0 S2 S3 and maybe we can introduce some kind of shorthand like this s of 2

8:44

3 4 0 2 3 which just more efficiently encodes the sequence of vertices which we mutated at

8:52

now actually this is expressed as a list but if you think about it for a little while you

8:58

start to realize that actually a different combinatorial object is really required here

9:03

we should really rather Express this as a pose set or a partially ordered set why because there was

9:13

not just a one-way role right there were there was some Forks in the road we had some choices

9:19

and that can beautifully be expressed by instead of writing a sort of a vertical list by rewriting

9:26

this sequence of mutations that we've done in this fashion here as a post set okay as a partial

9:34

ordered set where we're reading sort of from top to bottom so we're going down okay so if we're uh

9:42

you know potent These Bars here these edges sort of correspond to relations okay it's saying that

9:50

you know S2 is earlier than S3 if we're reading down and S2 is earlier than s0 but there's no

9:56

correspondence between s0 and S3 we can't say that one of these is is earlier than the other and what

10:02

we've been doing here when we looked at this thing here is we've really kind of gone through

10:06

this in a in a linear fashion choose S2 and then S3 and then S4 and then s0 and then S2 and then

10:14

S3 which respects or corresponds maybe it's better to say which respects the partial order

10:24

but there's other mutation sequence that also correspond to orderings of this post

10:29

set for example another one is this one we could start with S2 and then do s0 and then

10:35

do S3 and then do S2 and then do S4 and then do S3 you can check to see that that sort of

10:44

a different way of building up this maximal root population from the starting Singleton

10:52

but sort of going through a different sequence of mutations but the end result is the same

10:59

and in fact you can also have done it by going 2 2 0 3 4 2 3. so naturally what we're seeing

11:10

is that combinatorial structures emerge from the graphs the combinatorial structure is a post set

11:17

okay so we started with the D4 graph and this particular Singleton at the vertex that we call

11:25

vertex one and somehow the mutation game is giving us this post set this turns out to be a hugely

11:36

interesting and and Rich um relation that generates all kinds of just absolutely remarkable

11:43

and beautiful stuff it's just mind-boggling as we're going to see in subsequent videos.